Applications of Polynomial Systems

Cox, David

doi:10.1090/cbms/134

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Applications of Polynomial Systems

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David A. Cox, Amherst College, Amherst, MA

Publication: CBMS Regional Conference Series in Mathematics
Publication Year: 2020; Volume 134
ISBNs: 978-1-4704-5137-0 (print); 978-1-4704-5589-7 (online)
DOI: https://doi.org/10.1090/cbms/134

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J. Abbott, A. Bigatti, CoCoALib: a C++ library for doing Computations in Commutative Algebra, http://cocoa.dima.unige.it/cocoalib.
John Abbott, Anna Maria Bigatti, and Lorenzo Robbiano, Implicitization of hypersurfaces, J. Symbolic Comput. 81 (2017), 20–40. MR 3594322, DOI 10.1016/j.jsc.2016.11.002
S. Abhyankar, Polynomials and power series, Math. Intelligencer 3, Springer-Verlag, September 1972. Reprinted in Algebra, Arithmetic and Geometry with Applications (C. Christensen, G. Sundaram, A. Sathaye and C. Bajaj, eds.), Springer, New York, 2004, 783–784.
Shreeram S. Abhyankar, Historical ramblings in algebraic geometry and related algebra, Amer. Math. Monthly 83 (1976), no. 6, 409–448. MR 401754, DOI 10.2307/2318338
M. Adamer and M. Helmer, Steady state invariants and multistationarity for families of toric reaction networks, 2019, arXiv:1906.03931 [q-bio.MN].
Martin Aigner and Günter M. Ziegler, Proofs from The Book, 5th ed., Springer-Verlag, Berlin, 2014. Including illustrations by Karl H. Hofmann. MR 3288091, DOI 10.1007/978-3-662-44205-0
H. Alt, Über die Erzeugung gegebener ebener Kurven mit Hilfe des Gelenkvierseits, Z. angew. Math. Mech. 3 (1923), 13–19.
D. Arnold and C. Wampler, poster image created for the IMA thematic year on IMA Applications of Algebraic Geometry, https://www.ima.umn.edu/2006-2007.
Benjamin Assarf, Ewgenij Gawrilow, Katrin Herr, Michael Joswig, Benjamin Lorenz, Andreas Paffenholz, and Thomas Rehn, Computing convex hulls and counting integer points with polymake, Math. Program. Comput. 9 (2017), no. 1, 1–38. MR 3613012, DOI 10.1007/s12532-016-0104-z
L. Asimow and B. Roth, The rigidity of graphs, Trans. Amer. Math. Soc. 245 (1978), 279–289. MR 511410, DOI 10.1090/S0002-9947-1978-0511410-9
L. Asimow and B. Roth, The rigidity of graphs. II, J. Math. Anal. Appl. 68 (1979), no. 1, 171–190. MR 531431, DOI 10.1016/0022-247X(79)90108-2
W. Auzinger and H. J. Stetter, An elimination algorithm for the computation of all zeros of a system of multivariate polynomial equations, Numerical mathematics, Singapore 1988, Internat. Schriftenreihe Numer. Math., vol. 86, Birkhäuser, Basel, 1988, pp. 11–30. MR 1022943
C. Baciu and M. Kreuzer, Algebraisches Öl, Mitteilungen der DMV 19 (2011), 142–147.
Murad Banaji, Inheritance of oscillation in chemical reaction networks, Appl. Math. Comput. 325 (2018), 191–209. MR 3759136, DOI 10.1016/j.amc.2017.12.012
Murad Banaji and Casian Pantea, The inheritance of nondegenerate multistationarity in chemical reaction networks, SIAM J. Appl. Math. 78 (2018), no. 2, 1105–1130. MR 3784124, DOI 10.1137/16M1103506
E. Bartzos, I. Emiris, J. Legerský and E. Tsigaridas, On the maximal number of real embeddings of minimally rigid graphs in $\R ^2$, $\R ^3$ and $S^2$, J. Symbolic Comput. (2019), to appear, DOI 10.1016/j.jsc.2019.10.015.
I. G. Bashmakova and G. S. Smirnova, The beginnings and evolution of algebra, The Dolciani Mathematical Expositions, vol. 23, Mathematical Association of America, Washington, DC, 2000. Translated from the Russian by Abe Shenitzer with the editorial assistance of David A. Cox. MR 1735499
A. Baskar and S. Bandyopadhyay, An algorithm to compute the finite roots of large systems of polynomial equations arising in kinematic synthesis, Mech. Mach. Theory 133 (2019), 493–513.
Daniel J. Bates, Jonathan D. Hauenstein, Andrew J. Sommese, and Charles W. Wampler II, Adaptive multiprecision path tracking, SIAM J. Numer. Anal. 46 (2008), no. 2, 722–746. MR 2383209, DOI 10.1137/060658862
D. Bates, J. Hauenstein, A. Sommese and C. Wampler, Bertini: Software for Numerical Algebraic Geometry, https://bertini.nd.edu.
Daniel J. Bates, Jonathan D. Hauenstein, Andrew J. Sommese, and Charles W. Wampler, Numerically solving polynomial systems with Bertini, Software, Environments, and Tools, vol. 25, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2013. MR 3155500
Daniel J. Bates, Jonathan D. Hauenstein, Timothy M. McCoy, Chris Peterson, and Andrew J. Sommese, Recovering exact results from inexact numerical data in algebraic geometry, Exp. Math. 22 (2013), no. 1, 38–50. MR 3038781, DOI 10.1080/10586458.2013.737640
Carlos Beltrán and Luis Miguel Pardo, On Smale’s 17th problem: a probabilistic positive solution, Found. Comput. Math. 8 (2008), no. 1, 1–43. MR 2403529, DOI 10.1007/s10208-005-0211-0
Carlos Beltrán and Luis Miguel Pardo, Smale’s 17th problem: average polynomial time to compute affine and projective solutions, J. Amer. Math. Soc. 22 (2009), no. 2, 363–385. MR 2476778, DOI 10.1090/S0894-0347-08-00630-9
D. N. Bernstein, The number of roots of a system of equations, Funkcional. Anal. i Priložen. 9 (1975), no. 3, 1–4 (Russian). MR 0435072
S. Bernstein, Démonstration du théorème de Weierstrass fondée sur le calcul des probabilités, Comm. Kharkov Math. Soc. 13 (1912/13), 1–2.
E. Bézout, Sur le degré des équations résultantes de l’évanouissement des inconnues, Histoire de l’Académie Royale des Sciences (1764), 288–338.
Etienne Bézout, General theory of algebraic equations, Princeton University Press, Princeton, NJ, 2006. Translated from the 1779 French original by Eric Feron. MR 2215193
Frédéric Bihan and Alicia Dickenstein, Descartes’ rule of signs for polynomial systems supported on circuits, Int. Math. Res. Not. IMRN 22 (2017), 6867–6893. MR 3737324, DOI 10.1093/imrn/rnw199
F. Bihan, A. Dickenstein and M. Giaroli, Lower bounds for positive roots and regions of multistationarity in chemical reaction networks, 2018, arXiv:1807.05157[math.DS].
Frédéric Bihan, J. Maurice Rojas, and Frank Sottile, On the sharpness of fewnomial bounds and the number of components of fewnomial hypersurfaces, Algorithms in algebraic geometry, IMA Vol. Math. Appl., vol. 146, Springer, New York, 2008, pp. 15–20. MR 2397935, DOI 10.1007/978-0-387-75155-9_{2}
Frédéric Bihan, Francisco Santos, and Pierre-Jean Spaenlehauer, A polyhedral method for sparse systems with many positive solutions, SIAM J. Appl. Algebra Geom. 2 (2018), no. 4, 620–645. MR 3892410, DOI 10.1137/18M1181912
Lenore Blum, Felipe Cucker, Michael Shub, and Steve Smale, Complexity and real computation, Springer-Verlag, New York, 1998. With a foreword by Richard M. Karp. MR 1479636, DOI 10.1007/978-1-4612-0701-6
Leonard M. Blumenthal, Theory and applications of distance geometry, Oxford, at the Clarendon Press, 1953. MR 0054981
O. Bohigas, M. Manubens, and L. Ros, A linear relaxation method for computing workspace slices of the Stewart platform, J. Mechanisms Robotics 5 (2013), issue 1.
Ciprian Borcea and Ileana Streinu, The number of embeddings of minimally rigid graphs, Discrete Comput. Geom. 31 (2004), no. 2, 287–303. MR 2060642, DOI 10.1007/s00454-003-2902-0
Balázs Boros, Existence of positive steady states for weakly reversible mass-action systems, SIAM J. Math. Anal. 51 (2019), no. 1, 435–449. MR 3911717, DOI 10.1137/17M115534X
Jacob A. Boswell and Vivek Mukundan, Rees algebras and almost linearly presented ideals, J. Algebra 460 (2016), 102–127. MR 3510395, DOI 10.1016/j.jalgebra.2016.03.035
Nicolás Botbol, The implicit equation of a multigraded hypersurface, J. Algebra 348 (2011), 381–401. MR 2852248, DOI 10.1016/j.jalgebra.2011.09.019
Nicolás Botbol, Laurent Busé, and Marc Chardin, Fitting ideals and multiple points of surface parameterizations, J. Algebra 420 (2014), 486–508. MR 3261469, DOI 10.1016/j.jalgebra.2014.07.028
Nicolás Botbol and Alicia Dickenstein, Implicitization of rational hypersurfaces via linear syzygies: a practical overview, J. Symbolic Comput. 74 (2016), 493–512. MR 3424053, DOI 10.1016/j.jsc.2015.09.001
Nicolás Botbol, Alicia Dickenstein, and Marc Dohm, Matrix representations for toric parametrizations, Comput. Aided Geom. Design 26 (2009), no. 7, 757–771. MR 2569833, DOI 10.1016/j.cagd.2009.03.005
A. Brill, Ueber eine Eigenschaft der Resultante, Math. Ann. 16 (1880), no. 3, 345–347 (German). MR 1510029, DOI 10.1007/BF01447121
A. Brill, Elimination und Geometrie in den letzten Jahrzehnten, in Verhandlungen des dritten internationalen Mathematiker-Kongresses in Heidelberg 1904, Leipzig 1905, 275–283.
A. Brill and M. Noether, Die Entwicklung der Theorie der algebraischen Functionen in älterer und neuerer Zeit, Jahresbericht der Deutschen Mathematiker Vereinigung 3 (1892–1893), 111–566,
E. Brugallé, I. Itenberg, G. Mikhalkin and K. Shaw, Brief introduction to tropical geometry, in Proceedings of the Gökova (S. Akbulut, D. Auroux and T. Önder, eds.), International Press, Boston, 2016, 1–75.
B. Buchberger, Ein Algorithmus zum Auffinden der Basiselemente des Restklassenringes nach einem nulldimensionalen Polynomideal, Ph.D. Thesis, University of Innsbruck, 1965.
B. Buchberger, Ein algorithmisches Kriterium für die Lösbarkeit eines algebraischen Gleichungssystems, Aequationes Math. 4 (1970), 374–383 (German). MR 268178, DOI 10.1007/BF01844169
H. M. Möller and B. Buchberger, The construction of multivariate polynomials with preassigned zeros, Computer algebra (Marseille, 1982) Lecture Notes in Comput. Sci., vol. 144, Springer, Berlin-New York, 1982, pp. 24–31. MR 680050
Peter Bürgisser and Felipe Cucker, Condition, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 349, Springer, Heidelberg, 2013. The geometry of numerical algorithms. MR 3098452, DOI 10.1007/978-3-642-38896-5
Peter Bürgisser and Felipe Cucker, On a problem posed by Steve Smale, Ann. of Math. (2) 174 (2011), no. 3, 1785–1836. MR 2846491, DOI 10.4007/annals.2011.174.3.8
Laurent Busé, Residual resultant over the projective plane and the implicitization problem, Proceedings of the 2001 International Symposium on Symbolic and Algebraic Computation, ACM, New York, 2001, pp. 48–55. MR 2049730, DOI 10.1145/384101.384109
Laurent Busé, Resultants of determinantal varieties, J. Pure Appl. Algebra 193 (2004), no. 1-3, 71–97. MR 2076379, DOI 10.1016/j.jpaa.2004.02.010
Laurent Busé, On the equations of the moving curve ideal of a rational algebraic plane curve, J. Algebra 321 (2009), no. 8, 2317–2344. MR 2501523, DOI 10.1016/j.jalgebra.2009.01.030
Busé Laurent, Implicit matrix representations of rational Bézier curves and surfaces, Comput.-Aided Des. 46 (2014), 14–24. MR 3124172, DOI 10.1016/j.cad.2013.08.014
Laurent Busé and Thang Luu Ba, The surface/surface intersection problem by means of matrix based representations, Comput. Aided Geom. Design 29 (2012), no. 8, 579–598. MR 2948569, DOI 10.1016/j.cagd.2012.04.002
Laurent Busé and Marc Chardin, Implicitizing rational hypersurfaces using approximation complexes, J. Symbolic Comput. 40 (2005), no. 4-5, 1150–1168. MR 2172855, DOI 10.1016/j.jsc.2004.04.005
Laurent Busé, Marc Chardin, and Jean-Pierre Jouanolou, Torsion of the symmetric algebra and implicitization, Proc. Amer. Math. Soc. 137 (2009), no. 6, 1855–1865. MR 2480264, DOI 10.1090/S0002-9939-09-09550-1
Laurent Busé, Marc Chardin, and Aron Simis, Elimination and nonlinear equations of Rees algebras, J. Algebra 324 (2010), no. 6, 1314–1333. With an appendix in French by Joseph Oesterlé. MR 2671807, DOI 10.1016/j.jalgebra.2010.07.006
Laurent Busé, David Cox, and Carlos D’Andrea, Implicitization of surfaces in ${\Bbb P}^3$ in the presence of base points, J. Algebra Appl. 2 (2003), no. 2, 189–214. MR 1980408, DOI 10.1142/S0219498803000489
Laurent Busé and Carlos D’Andrea, Singular factors of rational plane curves, J. Algebra 357 (2012), 322–346. MR 2905259, DOI 10.1016/j.jalgebra.2012.01.030
Laurent Busé, Mohamed Elkadi, and Bernard Mourrain, Generalized resultants over unirational algebraic varieties, J. Symbolic Comput. 29 (2000), no. 4-5, 515–526. Symbolic computation in algebra, analysis, and geometry (Berkeley, CA, 1998). MR 1769653, DOI 10.1006/jsco.1999.0304
L. Busé, M. Elkadi, and B. Mourrain, Resultant over the residual of a complete intersection, J. Pure Appl. Algebra 164 (2001), no. 1-2, 35–57. Effective methods in algebraic geometry (Bath, 2000). MR 1854329, DOI 10.1016/S0022-4049(00)00144-4
Laurent Busé and Jean-Pierre Jouanolou, On the closed image of a rational map and the implicitization problem, J. Algebra 265 (2003), no. 1, 312–357. MR 1984914, DOI 10.1016/S0021-8693(03)00181-9
Laurent Busé and Anna Karasoulou, Resultant of an equivariant polynomial system with respect to the symmetric group, J. Symbolic Comput. 76 (2016), 142–157. MR 3461263, DOI 10.1016/j.jsc.2015.12.004
Jose Capco, Matteo Gallet, Georg Grasegger, Christoph Koutschan, Niels Lubbes, and Josef Schicho, The number of realizations of a Laman graph, SIAM J. Appl. Algebra Geom. 2 (2018), no. 1, 94–125. MR 3771397, DOI 10.1137/17M1118312
J. Capco, M. Gallet, G. Grasegger, C. Koutschan, N. Lubbes and J. Schicho, Computing the number of realizations of a Laman graph, Electronic Notes in Disc. Math. 61 (2017), 207–213. See \cite{Capco} for more details.
A. Cauchy, Recherches sur les polyèdres, Premier Mémoire, Journal de l’École Polytechnique 9 (1813), 68–86.
A. Cayley, A theorem in the geometry of position, Cambridge Math. Journal 2 (1841), 267–271.
A. Cayley, On linear transformations, Cambridge and Dublin Math. Journal 1 (1846), 104–122.
A. Cayley, Recherches sur l’élimination, et sur la théorie des courbes, J. Reine Angew. Math. 34 (1847), 30–45 (French). MR 1578563, DOI 10.1515/crll.1847.34.30
A. Cayley, Note sur la méthode d’élimination de Bezout, J. Reine Angew. Math. 53 (1857), 366–367 (French). MR 1579014, DOI 10.1515/crll.1857.53.366
A. Cayley, A fourth memoir upon quantics, Phil. Trans. 148 (1858), 415–427.
A. Cayley, Nouvelles recherches sur l’élimination et la théorie des courbes, J. Reine Angew. Math. 63 (1864), 34–39 (French). MR 1579252, DOI 10.1515/crll.1864.63.34
Marc Chardin, Implicitization using approximation complexes, Algebraic geometry and geometric modeling, Math. Vis., Springer, Berlin, 2006, pp. 23–35. MR 2279841, DOI 10.1007/978-3-540-33275-6_{2}
Falai Chen, David Cox, and Yang Liu, The $\mu$-basis and implicitization of a rational parametric surface, J. Symbolic Comput. 39 (2005), no. 6, 689–706. MR 2168614, DOI 10.1016/j.jsc.2005.01.003
F. Chen, L. Shen, and J. Deng, Implicitization and parametrization of quadratic and cubic surfaces by $\mu$-bases, Computing 79 (2007), no. 2-4, 131–142. MR 2295516, DOI 10.1007/s00607-006-0192-0
Tianran Chen, Tsung-Lin Lee, and Tien-Yien Li, Hom4PS-3: a parallel numerical solver for systems of polynomial equations based on polyhedral homotopy continuation methods, Mathematical software—ICMS 2014, Lecture Notes in Comput. Sci., vol. 8592, Springer, Heidelberg, 2014, pp. 183–190. MR 3334764, DOI 10.1007/978-3-662-44199-2_{3}0
B. Clarke, Stability of complex reaction networks, Adv. Chem. Phys. 42 (1980), 1–213.
C. Clement, A. Lee-St. John, and J. Sidman, Hyperbanana graphs, Proceedings of 25th Canadian Conference on Computational Geometry, (2013), paper 45, http://cccg.ca/proceedings/2013/.
Diego Cifuentes and Pablo A. Parrilo, Sampling algebraic varieties for sum of squares programs, SIAM J. Optim. 27 (2017), no. 4, 2381–2404. MR 3724261, DOI 10.1137/15M1052548
C. Conradi, E. Feliu and M. Mincheva, On the existence of Hopf bifurcations in the sequential and distributive double phosphorylation cycle, 2019, arXiv:1905.08129[q-bio.MN].
C. Conradi, E. Feliu, M. Mincheva and C. Wiuf, Identifying parameter regions for multistationarity, PLoS Comput. Biol. 13 (2017), e1005751.
C. Conradi, D. Flockerzi, J. Raisch and J. Stelling, Subnetwork analysis reveals dynamic features of complex (bio)chemical networks, PNAS 104 (2007), 19175–19180.
Carsten Conradi, Maya Mincheva, and Anne Shiu, Emergence of oscillations in a mixed-mechanism phosphorylation system, Bull. Math. Biol. 81 (2019), no. 6, 1829–1852. MR 3945912, DOI 10.1007/s11538-019-00580-6
Carsten Conradi and Casian Pantea, Multistationarity in biochemical networks: results, analysis, and examples, Algebraic and combinatorial computational biology, Math. Sci. Eng., Academic Press, London, 2019, pp. 279–317. MR 3839594
Carsten Conradi and Anne Shiu, A global convergence result for processive multisite phosphorylation systems, Bull. Math. Biol. 77 (2015), no. 1, 126–155. MR 3303108, DOI 10.1007/s11538-014-0054-4
C. Conradi and A. Shiu, Dynamics of post-translational modification systems: recent progress and future directions, Biophys. J. 114 (2018), 507–515.
Teresa Cortadellas Benítez and Carlos D’Andrea, Minimal generators of the defining ideal of the Rees algebra associated to monoid parameterizations, Comput. Aided Geom. Design 27 (2010), no. 6, 461–473. MR 2657547, DOI 10.1016/j.cagd.2010.04.003
Teresa Cortadellas Benítez and Carlos D’Andrea, Minimal generators of the defining ideal of the Rees algebra associated with a rational plane parametrization with $\mu =2$, Canad. J. Math. 66 (2014), no. 6, 1225–1249. MR 3270782, DOI 10.4153/CJM-2013-035-1
Teresa Cortadellas Benítez and Carlos D’Andrea, Rational plane curves parameterizable by conics, J. Algebra 373 (2013), 453–480. MR 2995038, DOI 10.1016/j.jalgebra.2012.09.034
Teresa Cortadellas Benítez, David A. Cox, and Carlos D’Andrea, The Rees algebra of parametric curves via liftings, J. Pure Appl. Algebra 224 (2020), no. 2, 869–893. MR 3987981, DOI 10.1016/j.jpaa.2019.06.015
M. Coste, An Introduction to Semialgebraic Geometry, Univ. of Rennes, 2002, http://gcomte.perso.math.cnrs.fr/M2/CosteIntroToSemialGeo.pdf.
David A. Cox, Equations of parametric curves and surfaces via syzygies, Symbolic computation: solving equations in algebra, geometry, and engineering (South Hadley, MA, 2000) Contemp. Math., vol. 286, Amer. Math. Soc., Providence, RI, 2001, pp. 1–20. MR 1874268, DOI 10.1090/conm/286/04751
David Cox, Curves, surfaces, and syzygies, Topics in algebraic geometry and geometric modeling, Contemp. Math., vol. 334, Amer. Math. Soc., Providence, RI, 2003, pp. 131–150. MR 2039970, DOI 10.1090/conm/334/05979
David A. Cox, The moving curve ideal and the Rees algebra, Theoret. Comput. Sci. 392 (2008), no. 1-3, 23–36. MR 2394983, DOI 10.1016/j.tcs.2007.10.012
David A. Cox, Solving equations via algebras, Solving polynomial equations, Algorithms Comput. Math., vol. 14, Springer, Berlin, 2005, pp. 63–123. MR 2161986, DOI 10.1007/3-540-27357-3_{2}
David Cox, What is a toric variety?, Topics in algebraic geometry and geometric modeling, Contemp. Math., vol. 334, Amer. Math. Soc., Providence, RI, 2003, pp. 203–223. MR 2039974, DOI 10.1090/conm/334/05983
D. Cox and P. Clarke, Moment maps, strict linear precision, and maximum likelihood degree one, 2018, arXiv:1810.03672 [math.AG].
David Cox, Ronald Goldman, and Ming Zhang, On the validity of implicitization by moving quadrics of rational surfaces with no base points, J. Symbolic Comput. 29 (2000), no. 3, 419–440. MR 1751389, DOI 10.1006/jsco.1999.0325
David Cox, J. William Hoffman, and Haohao Wang, Syzygies and the Rees algebra, J. Pure Appl. Algebra 212 (2008), no. 7, 1787–1796. MR 2400743, DOI 10.1016/j.jpaa.2007.11.006
David A. Cox and Anthony A. Iarrobino, Strata of rational space curves, Comput. Aided Geom. Design 32 (2015), 50–68. MR 3301274, DOI 10.1016/j.cagd.2014.11.004
David Cox, Andrew R. Kustin, Claudia Polini, and Bernd Ulrich, A study of singularities on rational curves via syzygies, Mem. Amer. Math. Soc. 222 (2013), no. 1045, x+116. MR 3059370, DOI 10.1090/S0065-9266-2012-00674-5
David A. Cox, John Little, and Donal O’Shea, Ideals, varieties, and algorithms, 4th ed., Undergraduate Texts in Mathematics, Springer, Cham, 2015. An introduction to computational algebraic geometry and commutative algebra. MR 3330490, DOI 10.1007/978-3-319-16721-3
David A. Cox, John Little, and Donal O’Shea, Using algebraic geometry, 2nd ed., Graduate Texts in Mathematics, vol. 185, Springer, New York, 2005. MR 2122859
David A. Cox, John B. Little, and Henry K. Schenck, Toric varieties, Graduate Studies in Mathematics, vol. 124, American Mathematical Society, Providence, RI, 2011. MR 2810322, DOI 10.1090/gsm/124
D. Cox, K.-N. Lin and G. Sosa, Multi-Rees algebras and toric dynamical systems, Proc. Amer. Math. Soc., to appear.
David A. Cox, Thomas W. Sederberg, and Falai Chen, The moving line ideal basis of planar rational curves, Comput. Aided Geom. Design 15 (1998), no. 8, 803–827. MR 1638732, DOI 10.1016/S0167-8396(98)00014-4
G. Craciun, Toric differential inclusions and a proof of the global attractor conjecture, 2015, arXiv:1501.02860 [math.DS].
Gheorghe Craciun, Polynomial dynamical systems, reaction networks, and toric differential inclusions, SIAM J. Appl. Algebra Geom. 3 (2019), no. 1, 87–106. MR 3920470, DOI 10.1137/17M1129076
Gheorghe Craciun and Martin Feinberg, Multiple equilibria in complex chemical reaction networks. I. The injectivity property, SIAM J. Appl. Math. 65 (2005), no. 5, 1526–1546. MR 2177713, DOI 10.1137/S0036139904440278
Gheorghe Craciun and Martin Feinberg, Multiple equilibria in complex chemical reaction networks: semiopen mass action systems, SIAM J. Appl. Math. 70 (2010), no. 6, 1859–1877. MR 2596505, DOI 10.1137/090756387
Gheorghe Craciun, Alicia Dickenstein, Anne Shiu, and Bernd Sturmfels, Toric dynamical systems, J. Symbolic Comput. 44 (2009), no. 11, 1551–1565. MR 2561288, DOI 10.1016/j.jsc.2008.08.006
Gheorghe Craciun, J. William Helton, and Ruth J. Williams, Homotopy methods for counting reaction network equilibria, Math. Biosci. 216 (2008), no. 2, 140–149. MR 2477000, DOI 10.1016/j.mbs.2008.09.001
Gheorghe Craciun, Luis David García-Puente, and Frank Sottile, Some geometrical aspects of control points for toric patches, Mathematical methods for curves and surfaces, Lecture Notes in Comput. Sci., vol. 5862, Springer, Berlin, 2010, pp. 111–135. MR 3193313, DOI 10.1007/978-3-642-11620-9_{9}
G. Cramer, Introduction à l’analyse des lignes courbes algébriques, Frères Cramer et Cl. Philibert, Genêve, 1750.
Noah S. Daleo and Jonathan D. Hauenstein, Numerically deciding the arithmetically Cohen-Macaulayness of a projective scheme, J. Symbolic Comput. 72 (2016), 128–146. MR 3369053, DOI 10.1016/j.jsc.2015.01.001
Carlos D’Andrea, Hoon Hong, Teresa Krick, and Agnes Szanto, An elementary proof of Sylvester’s double sums for subresultants, J. Symbolic Comput. 42 (2007), no. 3, 290–297. MR 2292636, DOI 10.1016/j.jsc.2006.09.003
Amit Khetan and Carlos D’Andrea, Implicitization of rational surfaces using toric varieties, J. Algebra 303 (2006), no. 2, 543–565. MR 2255122, DOI 10.1016/j.jalgebra.2005.05.028
Carlos D’Andrea and Martín Sombra, A Poisson formula for the sparse resultant, Proc. Lond. Math. Soc. (3) 110 (2015), no. 4, 932–964. MR 3335291, DOI 10.1112/plms/pdu069
W. Decker, G-M. Greuel, G. Pfister and H. Schönemann, Singular (4-1-1) — A computer algebra system for polynomial computations, 2018, http://www.singular.uni-kl.de.
W. L. F. Degen, The types of rational $(2,1)$-Bézier surfaces, Comput. Aided Geom. Design 16 (1999), no. 7, 639–648. Dedicated to Paul de Faget de Casteljau. MR 1718055, DOI 10.1016/S0167-8396(99)00028-X
P. De Leenheer, An elementary proof of a matrix tree theorem for directed graphs, 2019, arXiv:1904.12221 [math.CO].
Peter Deuflhard and Andreas Hohmann, Numerical analysis in modern scientific computing, 2nd ed., Texts in Applied Mathematics, vol. 43, Springer-Verlag, New York, 2003. An introduction. MR 1949263, DOI 10.1007/978-0-387-21584-6
Sandra Di Rocco, David Eklund, Chris Peterson, and Andrew J. Sommese, Chern numbers of smooth varieties via homotopy continuation and intersection theory, J. Symbolic Comput. 46 (2011), no. 1, 23–33. MR 2736356, DOI 10.1016/j.jsc.2010.06.026
A. Dickenstein, Algebra and geometry in the study of enzymatic cascades, in World Women in Mathematics 2018 – Proceedings of the First World Meeting for Women in Mathematics $(WM)^2$ (C. Araujo, G. Benkart, C. Praeger and B. Tanbay, eds.), Association for Women in Mathematics Series, Springer, 2019.
Alicia Dickenstein, Biochemical reaction networks: an invitation for algebraic geometers, Mathematical Congress of the Americas, Contemp. Math., vol. 656, Amer. Math. Soc., Providence, RI, 2016, pp. 65–83. MR 3457596, DOI 10.1090/conm/656/13076
Alicia Dickenstein and Ioannis Z. Emiris, Multihomogeneous resultant formulae by means of complexes, J. Symbolic Comput. 36 (2003), no. 3-4, 317–342. International Symposium on Symbolic and Algebraic Computation (ISSAC’2002) (Lille). MR 2004032, DOI 10.1016/S0747-7171(03)00086-5
A. Dickenstein and E. Feliu, Algebraic Methods for Biochemical Reaction Networks, in preparation.
Alicia Dickenstein, Eva Maria Feichtner, and Bernd Sturmfels, Tropical discriminants, J. Amer. Math. Soc. 20 (2007), no. 4, 1111–1133. MR 2328718, DOI 10.1090/S0894-0347-07-00562-0
Alicia Dickenstein, Mercedes Pérez Millán, Anne Shiu, and Xiaoxian Tang, Multistationarity in structured reaction networks, Bull. Math. Biol. 81 (2019), no. 5, 1527–1581. MR 3935185, DOI 10.1007/s11538-019-00572-6
Reinhard Diestel, Graph theory, 5th ed., Graduate Texts in Mathematics, vol. 173, Springer, Berlin, 2017. MR 3644391, DOI 10.1007/978-3-662-53622-3
A. L. Dixon, The Eliminant of Three Quantics in two Independent Variables, Proc. London Math. Soc. (2) 7 (1909), 49–69. MR 1575687, DOI 10.1112/plms/s2-7.1.49
Eliana Duarte and Hal Schenck, Tensor product surfaces and linear syzygies, Proc. Amer. Math. Soc. 144 (2016), no. 1, 65–72. MR 3415577, DOI 10.1090/proc/12703
E. Dufresne, P. Edwards, H. Harrington and J. Hauenstein, Sampling real algebraic varieties for topological data analysis, arXiv:1802.07716 [math.AT].
Jan Draisma, Emil Horobeţ, Giorgio Ottaviani, Bernd Sturmfels, and Rekha R. Thomas, The Euclidean distance degree of an algebraic variety, Found. Comput. Math. 16 (2016), no. 1, 99–149. MR 3451425, DOI 10.1007/s10208-014-9240-x
B. Edelstein, Biochemical models with multiple steady states and hysteresis, J. Theor. Biol. 29 (1970), 57–62.
W. Edge, The theory of ruled surfaces, Cambridge Univ, Press, 1931.
David Eisenbud, Commutative algebra, Graduate Texts in Mathematics, vol. 150, Springer-Verlag, New York, 1995. With a view toward algebraic geometry. MR 1322960, DOI 10.1007/978-1-4612-5350-1
David Eisenbud, The ReesAlgebra package in Macaulay2, J. Softw. Algebra Geom. 8 (2018), 49–60. MR 3857649, DOI 10.2140/jsag.2018.8.49
David Eisenbud and Bernd Sturmfels, Binomial ideals, Duke Math. J. 84 (1996), no. 1, 1–45. MR 1394747, DOI 10.1215/S0012-7094-96-08401-X
Mohamed Elkadi, André Galligo, and Thi Ha Lê, Parametrized surfaces in $\Bbb P^3$ of bidegree $(1,2)$, ISSAC 2004, ACM, New York, 2004, pp. 141–148. MR 2126936, DOI 10.1145/1005285.1005307
Federigo Enriques and Oscar Chisini, Lezioni sulla teoria geometrica delle equazioni e delle funzioni algebriche. 1. Vol. I, II, Collana di Matematica [Mathematics Collection], vol. 5, Zanichelli Editore S.p.A., Bologna, 1985 (Italian). Reprint of the 1915 and 1918 editions. MR 966664
James Farre, Helena Kleinschmidt, Jessica Sidman, Audrey St. John, Stephanie Stark, Louis Theran, and Xilin Yu, Algorithms for detecting dependencies and rigid subsystems for CAD, Comput. Aided Geom. Design 47 (2016), 130–149. MR 3545987, DOI 10.1016/j.cagd.2016.06.001
Jean-Charles Faugère, Guillaume Moroz, Fabrice Rouillier, and Mohab Safey El Din, Classification of the perspective-three-point problem, discriminant variety and real solving polynomial systems of inequalities, ISSAC 2008, ACM, New York, 2008, pp. 79–86. MR 2500376, DOI 10.1145/1390768.1390782
Ivor D. Faux and Michael J. Pratt, Computational geometry for design and manufacture, Mathematics and its Applications, Ellis Horwood Ltd., Chichester; Halsted Press [John Wiley & Sons], New York-Chichester-Brisbane, 1979. MR 524724
Martin Feinberg, Complex balancing in general kinetic systems, Arch. Rational Mech. Anal. 49 (1972/73), 187–194. MR 413930, DOI 10.1007/BF00255665
M. Feinberg, Lectures On Chemical Reaction Networks, 1979, https://crnt.osu.edu/LecturesOnReactionNetworks.
Martin Feinberg and F. J. M. Horn, Chemical mechanism structure and the coincidence of the stoichiometric and kinetic subspaces, Arch. Rational. Mech. Anal. 66 (1977), no. 1, 83–97. MR 0446109, DOI 10.1007/BF00250853
E. Feliu, A. Rendall and C. Wiuf, A proof of unlimited multistability for phosphorylation cycles, 2019, arXiv:1904.02983[q-bio.MN].
Elisenda Feliu and Carsten Wiuf, Variable elimination in chemical reaction networks with mass-action kinetics, SIAM J. Appl. Math. 72 (2012), no. 4, 959–981. MR 2968758, DOI 10.1137/110847305
E. Feliu and C. Wiuf, Enzyme-sharing as a cause of multi-stationarity in signalling systems, J. R. Soc. Interface 9 (2011), 1224–1232.
E. Feliu and C. Wiuf, Simplifying biochemical models with intermediate species, J. R. Soc. Interface 10 (2013), 20130484.
H. Ferguson and D. Bailey, A polynomial time, numerically stable integer relation algorithm, RNR Techn. Rept. (1992), RNR-91-032.
Dietrich Flockerzi, Katharina Holstein, and Carsten Conradi, $n$-site phosphorylation systems with $2n-1$ steady states, Bull. Math. Biol. 76 (2014), no. 8, 1892–1916. MR 3255149, DOI 10.1007/s11538-014-9984-0
Louiza Fouli and Kuei-Nuan Lin, Rees algebras of square-free monomial ideals, J. Commut. Algebra 7 (2015), no. 1, 25–54. MR 3316984, DOI 10.1216/JCA-2015-7-1-25
William Fulton, Introduction to toric varieties, Annals of Mathematics Studies, vol. 131, Princeton University Press, Princeton, NJ, 1993. The William H. Roever Lectures in Geometry. MR 1234037, DOI 10.1515/9781400882526
Thi-Ha Lê and André Galligo, General classification of $(1,2)$ parametric surfaces in $\Bbb P^3$, Geometric modeling and algebraic geometry, Springer, Berlin, 2008, pp. 93–113. MR 2381606, DOI 10.1007/978-3-540-72185-7_{6}
Luis David Garcia-Puente and Frank Sottile, Linear precision for parametric patches, Adv. Comput. Math. 33 (2010), no. 2, 191–214. MR 2659586, DOI 10.1007/s10444-009-9126-7
L. García-Puente, F. Sottile and C. Zhu, Toric degenerations of Bézier patches, ACM Transactions on Graphics 30 (2011), 110:1–110:10.
Karin Gatermann, Counting stable solutions of sparse polynomial systems in chemistry, Symbolic computation: solving equations in algebra, geometry, and engineering (South Hadley, MA, 2000) Contemp. Math., vol. 286, Amer. Math. Soc., Providence, RI, 2001, pp. 53–69. MR 1874271, DOI 10.1090/conm/286/04754
Karin Gatermann and Birkett Huber, A family of sparse polynomial systems arising in chemical reaction systems, J. Symbolic Comput. 33 (2002), no. 3, 275–305. MR 1882230, DOI 10.1006/jsco.2001.0512
Karin Gatermann and Matthias Wolfrum, Bernstein’s second theorem and Viro’s method for sparse polynomial systems in chemistry, Adv. in Appl. Math. 34 (2005), no. 2, 252–294. MR 2110552, DOI 10.1016/j.aam.2004.04.003
Karin Gatermann, Markus Eiswirth, and Anke Sensse, Toric ideals and graph theory to analyze Hopf bifurcations in mass action systems, J. Symbolic Comput. 40 (2005), no. 6, 1361–1382. MR 2178092, DOI 10.1016/j.jsc.2005.07.002
I. M. Gel′fand, M. M. Kapranov, and A. V. Zelevinsky, Discriminants, resultants, and multidimensional determinants, Mathematics: Theory & Applications, Birkhäuser Boston, Inc., Boston, MA, 1994. MR 1264417, DOI 10.1007/978-0-8176-4771-1
G. Giambelli, Sulle varietà rappresentate coll’annullare determinanti minori contenuti in un determinante simmetrico od emisimmetrico generico di forme, Atti R. Accad. Sci. Torino 44 (1905/1906), 102–125.
Magalí Giaroli, Frédéric Bihan, and Alicia Dickenstein, Regions of multistationarity in cascades of Goldbeter-Koshland loops, J. Math. Biol. 78 (2019), no. 4, 1115–1145. MR 3922967, DOI 10.1007/s00285-018-1304-0
M. Giaroli, R. Rischter, M. Pérez Millán and A. Dickenstein, Parameter regions that give rise to $2[n/2]+1$ positive steady states in the $n$-site phosphorylation system, 2019, arXiv:1904.11633[q-bio.MN].
Jean Giraud, Géométrie algébrique élémentaire, Publications Mathématiques d’Orsay, No. 77-75, Université de Paris-Sud, Département de Mathématiques, Orsay, 1977 (French). Cours de troisième cycle (1975–76). MR 0491670
Marc Giusti, Joos Heintz, and Juan Sabia, On the efficiency of effective Nullstellensätze, Comput. Complexity 3 (1993), no. 1, 56–95. MR 1220078, DOI 10.1007/BF01200407
M. Giusti, J. Heintz, J. E. Morais, J. Morgenstern, and L. M. Pardo, Straight-line programs in geometric elimination theory, J. Pure Appl. Algebra 124 (1998), no. 1-3, 101–146. MR 1600277, DOI 10.1016/S0022-4049(96)00099-0
Herman Gluck, Almost all simply connected closed surfaces are rigid, Geometric topology (Proc. Conf., Park City, Utah, 1974) Lecture Notes in Math., Vol. 438, Springer, Berlin, 1975, pp. 225–239. MR 0400239
Gilles Gnacadja, Reachability, persistence, and constructive chemical reaction networks (part II): a formalism for species composition in chemical reaction network theory and application to persistence, J. Math. Chem. 49 (2011), no. 10, 2137–2157. MR 2846708, DOI 10.1007/s10910-011-9896-2
Gilles Gnacadja, Reachability, persistence, and constructive chemical reaction networks (part III): a mathematical formalism for binary enzymatic networks and application to persistence, J. Math. Chem. 49 (2011), no. 10, 2158–2176. MR 2846709, DOI 10.1007/s10910-011-9895-3
Gene H. Golub and Charles F. Van Loan, Matrix computations, 4th ed., Johns Hopkins Studies in the Mathematical Sciences, Johns Hopkins University Press, Baltimore, MD, 2013. MR 3024913
Ron Goldman, Polar forms in geometric modeling and algebraic geometry, Topics in algebraic geometry and geometric modeling, Contemp. Math., vol. 334, Amer. Math. Soc., Providence, RI, 2003, pp. 3–24. MR 2039963, DOI 10.1090/conm/334/05972
D. Grayson and M. Stillman, Macaulay2, a software system for research in algebraic geometry, http://www.math.uiuc.edu/Macaulay2/.
Elizabeth Gross, Heather A. Harrington, Zvi Rosen, and Bernd Sturmfels, Algebraic systems biology: a case study for the Wnt pathway, Bull. Math. Biol. 78 (2016), no. 1, 21–51. MR 3452313, DOI 10.1007/s11538-015-0125-1
E. Gross, B. Davis, K. Ho, D. Bates, and H. Harrington, Numerical algebraic geometry for model selection and its application to the life sciences, J. R. Soc. Interface 13 (2016), 20160256.
Elizabeth Gross and Seth Sullivant, The maximum likelihood threshold of a graph, Bernoulli 24 (2018), no. 1, 386–407. MR 3706762, DOI 10.3150/16-BEJ881
A. Grothendieck and J. Dieudonné, Éléments de géométrie algébrique I-IV, Publications Mathématiques de l’IHÉS 4, 8, 11, 17, 20, 24, 28, 32 (1960-67). Second edition of EGA I, Springer, New York, 1971.
J. Gunawardena, Chemical reaction network theory for in-silico biologists, lecture notes, Harvard University, 2003, http://vcp.med.harvard.edu/papers/crnt.pdf.
J. Gunawardena, Theory and mathematical methods, Comprehensive Biophysics 9 (2012), 243–265.
Ruth Haas, Characterizations of arboricity of graphs, Ars Combin. 63 (2002), 129–137. MR 1898221
Kirk Haller, Audrey Lee-St.John, Meera Sitharam, Ileana Streinu, and Neil White, Body-and-cad geometric constraint systems, Comput. Geom. 45 (2012), no. 8, 385–405. MR 2922664, DOI 10.1016/j.comgeo.2010.06.003
H. Harrington, D. Mehta, H. Byrne and J. Hauenstein, Decomposing the parameter space of biological networks via a numerical discriminant approach, 2016, arXiv:1604.02623[q-bio.MN].
Joe Harris, Galois groups of enumerative problems, Duke Math. J. 46 (1979), no. 4, 685–724. MR 552521
Joe Harris and Loring W. Tu, On symmetric and skew-symmetric determinantal varieties, Topology 23 (1984), no. 1, 71–84. MR 721453, DOI 10.1016/0040-9383(84)90026-0
V. Hárs and J. Tóth, On the inverse problem of reaction kinetics, in Qualitative Theory of Differential Equations (M. Farkas, ed.), Colloquia Mathematica Societatis János Bolyai 30, North-Holland Publ. Co., Amsterdam, 1981, 363–379.
Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. MR 0463157
Jonathan D. Hauenstein, Numerically computing real points on algebraic sets, Acta Appl. Math. 125 (2013), 105–119. MR 3048642, DOI 10.1007/s10440-012-9782-3
J. Hauenstein and S. Sherman, Using monodromy to statistically estimate the number of solutions, in preparation.
Jonathan D. Hauenstein and Andrew J. Sommese, Membership tests for images of algebraic sets by linear projections, Appl. Math. Comput. 219 (2013), no. 12, 6809–6818. MR 3027848, DOI 10.1016/j.amc.2012.12.060
Jonathan D. Hauenstein and Andrew J. Sommese, What is numerical algebraic geometry? [Foreword]. part 3, J. Symbolic Comput. 79 (2017), no. part 3, 499–507. MR 3563094, DOI 10.1016/j.jsc.2016.07.015
Jonathan D. Hauenstein and Andrew J. Sommese, Witness sets of projections, Appl. Math. Comput. 217 (2010), no. 7, 3349–3354. MR 2733776, DOI 10.1016/j.amc.2010.08.067
Jonathan D. Hauenstein and Frank Sottile, Algorithm 921: alphaCertified: certifying solutions to polynomial systems, ACM Trans. Math. Software 38 (2012), no. 4, Art. 28, 20. MR 2972672, DOI 10.1145/2331130.2331136
Jonathan D. Hauenstein, Christian Ikenmeyer, and J. M. Landsberg, Equations for lower bounds on border rank, Exp. Math. 22 (2013), no. 4, 372–383. MR 3171099, DOI 10.1080/10586458.2013.825892
Joos Heintz, Bart Kuijpers, and Andrés Rojas Paredes, On the intrinsic complexity of elimination problems in effective algebraic geometry, Recent advances in real complexity and computation, Contemp. Math., vol. 604, Amer. Math. Soc., Providence, RI, 2013, pp. 129–150. MR 3204156, DOI 10.1090/conm/604/12071
Daniel Heldt, Martin Kreuzer, Sebastian Pokutta, and Hennie Poulisse, Approximate computation of zero-dimensional polynomial ideals, J. Symbolic Comput. 44 (2009), no. 11, 1566–1591. MR 2561289, DOI 10.1016/j.jsc.2008.11.010
Juliette Hell and Alan D. Rendall, Sustained oscillations in the MAP kinase cascade, Math. Biosci. 282 (2016), 162–173. MR 3573178, DOI 10.1016/j.mbs.2016.10.011
Uwe Helmke and John B. Moore, Optimization and dynamical systems, Communications and Control Engineering Series, Springer-Verlag London, Ltd., London, 1994. With a foreword by R. Brockett. MR 1299725, DOI 10.1007/978-1-4471-3467-1
J. Herzog, A. Simis, and W. V. Vasconcelos, Approximation complexes of blowing-up rings, J. Algebra 74 (1982), no. 2, 466–493. MR 647249, DOI 10.1016/0021-8693(82)90034-5
J. Herzog, A. Simis, and W. V. Vasconcelos, Approximation complexes of blowing-up rings. II, J. Algebra 82 (1983), no. 1, 53–83. MR 701036, DOI 10.1016/0021-8693(83)90173-4
Otto Hesse, Über die Bildung der Endgleichung, welche durch Elimination einer Variabeln aus zwei algebraischen Gleichungen hervorgeht, und die Bestimmung ihres Grades, J. Reine Angew. Math. 27 (1844), 1–5 (German). MR 1578380, DOI 10.1515/crll.1844.27.1
David Hilbert, Ueber die Theorie der algebraischen Formen, Math. Ann. 36 (1890), no. 4, 473–534 (German). MR 1510634, DOI 10.1007/BF01208503
David Hilbert, Ueber die vollen Invariantensysteme, Math. Ann. 42 (1893), no. 3, 313–373 (German). MR 1510781, DOI 10.1007/BF01444162
D. Hilbert, Mathematische Probleme, Nachrichten von der Königl. Gesellschaft der Wiss. zu Göttingen, mathematische-physkialische Klasse (1900), 253–297. English translation by M. Newson in Mathematical Developments Arising from Hilbert Problems, Proc. Symp. in Pure Math. XXVIII, Part 1, AMS, Providence, RI, 1976, 1–35.
M. Hochster, Rings of invariants of tori, Cohen-Macaulay rings generated by monomials, and polytopes, Ann. of Math. (2) 96 (1972), 318–337. MR 304376, DOI 10.2307/1970791
Katharina Holstein, Dietrich Flockerzi, and Carsten Conradi, Multistationarity in sequential distributed multisite phosphorylation networks, Bull. Math. Biol. 75 (2013), no. 11, 2028–2058. MR 3119394, DOI 10.1007/s11538-013-9878-6
Jooyoun Hong, Aron Simis, and Wolmer V. Vasconcelos, On the homology of two-dimensional elimination, J. Symbolic Comput. 43 (2008), no. 4, 275–292. MR 2402032, DOI 10.1016/j.jsc.2007.10.010
F. Horn, Necessary and sufficient conditions for complex balancing in chemical kinetics, Arch. Rational Mech. Anal. 49 (1972/73), 172–186. MR 413929, DOI 10.1007/BF00255664
F. Horn, The dynamics of open reaction systems, Mathematical aspects of chemical and biochemical problems and quantum chemistry (Proc. SIAM-AMS Sympos. Appl. Math., New York, 1974) SIAM-AMS Proceedings, Vol. VIII, Amer. Math. Soc., Providence, R.I., 1974, pp. 125–137. MR 0468667
F. Horn and R. Jackson, General mass action kinetics, Arch. Rational Mech. Anal. 47 (1972), 81–116. MR 400923, DOI 10.1007/BF00251225
Serkan Hoşten and Suela Ruffa, Introductory notes to algebraic statistics, Rend. Istit. Mat. Univ. Trieste 37 (2005), no. 1-2, 39–70 (2006). MR 2227048
June Huh, Varieties with maximum likelihood degree one, J. Algebr. Stat. 5 (2014), no. 1, 1–17. MR 3279951, DOI 10.18409/jas.v5i1.22
Craig Huneke and Bernd Ulrich, Divisor class groups and deformations, Amer. J. Math. 107 (1985), no. 6, 1265–1303 (1986). MR 815763, DOI 10.2307/2374407
W. Isaacson, Leonardo da Vinci, Simon & Schuster, New York, 2017.
C. G. J. Jacobi, Theoremata nova algebraica circa systema duarum aequationum, inter duas variabiles propositarum, J. Reine Angew. Math. 14 (1835), 281–288 (Latin). MR 1578080, DOI 10.1515/crll.1835.14.281
Bill Jackson and J. C. Owen, Point-line frameworks, Handbook of geometric constraint systems principles, Discrete Math. Appl. (Boca Raton), CRC Press, Boca Raton, FL, 2019, pp. 487–504. MR 3837067
Anders Jensen and Josephine Yu, Computing tropical resultants, J. Algebra 387 (2013), 287–319. MR 3056698, DOI 10.1016/j.jalgebra.2013.03.031
Gabriela Jeronimo, Teresa Krick, Juan Sabia, and Martín Sombra, The computational complexity of the Chow form, Found. Comput. Math. 4 (2004), no. 1, 41–117. MR 2035410, DOI 10.1007/s10208-002-0078-2
Gabriela Jeronimo and Juan Sabia, Sparse resultants and straight-line programs, J. Symbolic Comput. 87 (2018), 14–27. MR 3744337, DOI 10.1016/j.jsc.2017.05.005
Badal Joshi and Anne Shiu, Which small reaction networks are multistationary?, SIAM J. Appl. Dyn. Syst. 16 (2017), no. 2, 802–833. MR 3633777, DOI 10.1137/16M1069705
J. P. Jouanolou, Singularités rationnelles du résultant, Algebraic geometry (Proc. Summer Meeting, Univ. Copenhagen, Copenhagen, 1978) Lecture Notes in Math., vol. 732, Springer, Berlin, 1979, pp. 183–213 (French). MR 555699
J. P. Jouanolou, Idéaux résultants, Adv. in Math. 37 (1980), no. 3, 212–238 (French). MR 591727, DOI 10.1016/0001-8708(80)90034-1
J.-P. Jouanolou, Le formalisme du résultant, Adv. Math. 90 (1991), no. 2, 117–263 (French). MR 1142904, DOI 10.1016/0001-8708(91)90031-2
Jean-Pierre Jouanolou, Aspects invariants de l’élimination, Adv. Math. 114 (1995), no. 1, 1–174 (French). MR 1344713, DOI 10.1006/aima.1995.1042
Jean-Pierre Jouanolou, Résultant anisotrope, compléments et applications, Electron. J. Combin. 3 (1996), no. 2, Research Paper 2, approx. 91 (French). The Foata Festschrift. MR 1392487, DOI 10.37236/1260
J. P. Jouanolou, Formes d’inertie et résultant: un formulaire, Adv. Math. 126 (1997), no. 2, 119–250 (French). MR 1442307, DOI 10.1006/aima.1996.1609
Tadeusz Józefiak, Alain Lascoux, and Piotr Pragacz, Classes of determinantal varieties associated with symmetric and antisymmetric matrices, Proceedings of the Week of Algebraic Geometry (Bucharest, 1980) Teubner-Texte zur Mathematik, vol. 40, Teubner, Leipzig, 1981, pp. 106–108. MR 712519
M. M. Kapranov, B. Sturmfels, and A. V. Zelevinsky, Chow polytopes and general resultants, Duke Math. J. 67 (1992), no. 1, 189–218. MR 1174606, DOI 10.1215/S0012-7094-92-06707-X
Achim Kehrein, Martin Kreuzer, and Lorenzo Robbiano, An algebraist’s view on border bases, Solving polynomial equations, Algorithms Comput. Math., vol. 14, Springer, Berlin, 2005, pp. 169–202. MR 2161988, DOI 10.1007/3-540-27357-3_{4}
A. G. Khovanskiĭ, Fewnomials, Translations of Mathematical Monographs, vol. 88, American Mathematical Society, Providence, RI, 1991. Translated from the Russian by Smilka Zdravkovska. MR 1108621, DOI 10.1090/mmono/088
A. G. Khovanskii and Leonid Monin, The resultant of developed systems of Laurent polynomials, Mosc. Math. J. 17 (2017), no. 4, 717–740. MR 3734660, DOI 10.17323/1609-4514-2016-16-4-717-740
Csaba Király and Shin-ichi Tanigawa, Rigidity of body-bar-hinge frameworks, Handbook of geometric constraint systems principles, Discrete Math. Appl. (Boca Raton), CRC Press, Boca Raton, FL, 2019, pp. 435–459. MR 3837065
Steven L. Kleiman, Bertini and his two fundamental theorems, Rend. Circ. Mat. Palermo (2) Suppl. 55 (1998), 9–37. Studies in the history of modern mathematics, III. MR 1661859
Dimitra Kosta and Kaie Kubjas, Maximum likelihood estimation of symmetric group-based models via numerical algebraic geometry, Bull. Math. Biol. 81 (2019), no. 2, 337–360. MR 3902904, DOI 10.1007/s11538-018-0523-2
Rimvydas Krasauskas, Toric surface patches, Adv. Comput. Math. 17 (2002), no. 1-2, 89–113. Advances in geometrical algorithms and representations. MR 1902537, DOI 10.1023/A:1015289823859
Rimvydas Krasauskas and Ron Goldman, Toric Bézier patches with depth, Topics in algebraic geometry and geometric modeling, Contemp. Math., vol. 334, Amer. Math. Soc., Providence, RI, 2003, pp. 65–91. MR 2039967, DOI 10.1090/conm/334/05976
Martin Kreuzer and Henk Poulisse, Subideal border bases, Math. Comp. 80 (2011), no. 274, 1135–1154. MR 2772116, DOI 10.1090/S0025-5718-2010-02432-9
Martin Kreuzer and Lorenzo Robbiano, Computational linear and commutative algebra, Springer, Cham, 2016. MR 3559741, DOI 10.1007/978-3-319-43601-2
Martin Kreuzer, Hennie Poulisse, and Lorenzo Robbiano, From oil fields to Hilbert schemes, Approximate commutative algebra, Texts Monogr. Symbol. Comput., SpringerWienNewYork, Vienna, 2009, pp. 1–54. MR 2641155, DOI 10.1007/978-3-211-99314-9_{1}
Teresa Krick, Straight-line programs in polynomial equation solving, Foundations of computational mathematics: Minneapolis, 2002, London Math. Soc. Lecture Note Ser., vol. 312, Cambridge Univ. Press, Cambridge, 2004, pp. 96–136. MR 2189629
L. Kronecker, Grundzüge einer arithmetischen Theorie der algebraische Grössen, J. Reine Angew. Math. 92 (1882), 1–122 (German). MR 1579896, DOI 10.1515/crll.1882.92.1
Ernst Kunz, Residues and duality for projective algebraic varieties, University Lecture Series, vol. 47, American Mathematical Society, Providence, RI, 2008. With the assistance of and contributions by David A. Cox and Alicia Dickenstein. MR 2464546, DOI 10.1090/ulect/047
Andrew R. Kustin, The minimal free resolutions of the Huneke-Ulrich deviation two Gorenstein ideals, J. Algebra 100 (1986), no. 1, 265–304. MR 839583, DOI 10.1016/0021-8693(86)90078-5
Andrew Kustin, Claudia Polini, and Bernd Ulrich, The bi-graded structure of symmetric algebras with applications to Rees rings, J. Algebra 469 (2017), 188–250. MR 3563012, DOI 10.1016/j.jalgebra.2016.08.014
Andrew R. Kustin, Claudia Polini, and Bernd Ulrich, The equations defining blowup algebras of height three Gorenstein ideals, Algebra Number Theory 11 (2017), no. 7, 1489–1525. MR 3697146, DOI 10.2140/ant.2017.11.1489
Pierre Lairez, A deterministic algorithm to compute approximate roots of polynomial systems in polynomial average time, Found. Comput. Math. 17 (2017), no. 5, 1265–1292. MR 3709332, DOI 10.1007/s10208-016-9319-7
G. Laman, On graphs and rigidity of plane skeletal structures, J. Engrg. Math. 4 (1970), 331–340. MR 269535, DOI 10.1007/BF01534980
J. M. Landsberg, Geometry and complexity theory, Cambridge Studies in Advanced Mathematics, vol. 169, Cambridge University Press, Cambridge, 2017. MR 3729273, DOI 10.1017/9781108183192
Daniel Lazard, Algèbre linéaire sur $K[X_{1},\cdots ,X_{n}]$, et élimination, Bull. Soc. Math. France 105 (1977), no. 2, 165–190 (French). MR 491702
Daniel Lazard, Résolution des systèmes d’équations algébriques, Theoret. Comput. Sci. 15 (1981), no. 1, 77–110 (French, with English summary). MR 619687, DOI 10.1016/0304-3975(81)90064-5
Audrey Lee-St.John and Jessica Sidman, Combinatorics and the rigidity of CAD systems, Comput.-Aided Des. 45 (2013), no. 2, 473–482. MR 3041222, DOI 10.1016/j.cad.2012.10.030
G. Lecerf, Kronecker: a Magma package for polynomial system solving, lecerf.perso.math.cnrs.fr/software/kronecker/distribution.html.
A. K. Lenstra, H. W. Lenstra Jr., and L. Lovász, Factoring polynomials with rational coefficients, Math. Ann. 261 (1982), no. 4, 515–534. MR 682664, DOI 10.1007/BF01457454
J. Limbeck, Computation of Approximate Border Bases and Applications, PhD thesis, University of Passau, 2014.
Anton Leykin, Numerical algebraic geometry, J. Softw. Algebra Geom. 3 (2011), 5–10. MR 2881262, DOI 10.2140/jsag.2011.3.5
Kuei-Nuan Lin, Cohen-Macaulayness of Rees algebras of diagonal ideals, J. Commut. Algebra 6 (2014), no. 4, 561–586. MR 3294862, DOI 10.1216/JCA-2014-6-4-561
Kuei-Nuan Lin and Claudia Polini, Rees algebras of truncations of complete intersections, J. Algebra 410 (2014), 36–52. MR 3201047, DOI 10.1016/j.jalgebra.2014.03.022
D. Lindsey, Digital Gehry, Birkhäuser, Basel, 2001.
L. Lovász and Y. Yemini, On generic rigidity in the plane, SIAM J. Algebraic Discrete Methods 3 (1982), no. 1, 91–98. MR 644960, DOI 10.1137/0603009
T. Luu Ba, L. Busé and B. Mourrain, Curve/surface intersection problems by means of matrix representations, in Proceedings of the 2009 conference on Symbolic Numeric Computation (H. Kai and H. Sekigawa, eds.), ACM, New York, 2009, 71–78.
F. S. MacAulay, Some Formulae in Elimination, Proc. Lond. Math. Soc. 35 (1903), 3–27. MR 1577000, DOI 10.1112/plms/s1-35.1.3
F. Macaulay, The Algebraic Theory of Modular Systems, Cambridge Univ. Press, Cambridge, 1916.
Diane Maclagan and Bernd Sturmfels, Introduction to tropical geometry, Graduate Studies in Mathematics, vol. 161, American Mathematical Society, Providence, RI, 2015. MR 3287221, DOI 10.1090/gsm/161
J. Madsen, Equations of Rees algebras of ideals in two variables, arXiv:1511.04073[math.AC].
J. C. Maxwell. On the calculation of the equilibrium and stiffness of frames Philos. Mag. 27 (1864), 294–299.
H. Melenk, H. Möller and W. Neun, Symbolic solution of large stationary chemical kinetics problems, IMPACT Comput. Sci. Eng. 1 (1989), 138–167.
Karl Menger, New Foundation of Euclidean Geometry, Amer. J. Math. 53 (1931), no. 4, 721–745. MR 1506850, DOI 10.2307/2371222
F. Mertens, Über die bestimmenden Eigenschaften der Resultante von $n$ Formen mit $n$ Veränderlichen, Sitzungsberichte der Mathematisch-Naturwissenschaftlichen Classe der Kaiserlichen Akademie der Wissenschaften, II. Abtheilung 93 (1886), 527–566.
F. Mertens, Zur Theorie der Elimination, Teil II, Sitzungsber. Akad. Wien 108 (1899), 1344–1386.
Franz Meyer, Zur Theorie der reducibeln ganzen Functionen von $n$ Variabeln, Math. Ann. 30 (1887), no. 1, 30–74 (German). MR 1510432, DOI 10.1007/BF01564528
Ezra Miller and Bernd Sturmfels, Combinatorial commutative algebra, Graduate Texts in Mathematics, vol. 227, Springer-Verlag, New York, 2005. MR 2110098
Ferd. Minding, Ueber die Bestimmung des Grades einer durch Elimination hervorgehenden Gleichung, J. Reine Angew. Math. 22 (1841), 178–183 (German). MR 1578276, DOI 10.1515/crll.1841.22.178
Susan Morey and Bernd Ulrich, Rees algebras of ideals with low codimension, Proc. Amer. Math. Soc. 124 (1996), no. 12, 3653–3661. MR 1343713, DOI 10.1090/S0002-9939-96-03470-3
Stefan Müller, Elisenda Feliu, Georg Regensburger, Carsten Conradi, Anne Shiu, and Alicia Dickenstein, Sign conditions for injectivity of generalized polynomial maps with applications to chemical reaction networks and real algebraic geometry, Found. Comput. Math. 16 (2016), no. 1, 69–97. MR 3451424, DOI 10.1007/s10208-014-9239-3
Stefan Müller and Georg Regensburger, Generalized mass action systems: complex balancing equilibria and sign vectors of the stoichiometric and kinetic-order subspaces, SIAM J. Appl. Math. 72 (2012), no. 6, 1926–1947. MR 3024168, DOI 10.1137/110847056
Stefan Müller, Josef Hofbauer, and Georg Regensburger, On the bijectivity of families of exponential/generalized polynomial maps, SIAM J. Appl. Algebra Geom. 3 (2019), no. 3, 412–438. MR 3992044, DOI 10.1137/18M1178153
E. Netto, Vorlesungen über Algebra, Volume II, Teubner, Leipzig, 1900.
E. Netto and R. Le Vavasseur, Fonctions rationnelles, in Encyclopédie des sciences mathématiques pures et appliquées Tome I, Volume 2 (J. Molk, ed.), Gauthiers-Villars, Paris and B. G. Teubner, Leipzig, 1909, 1–232.
Isaac Newton, The mathematical papers of Isaac Newton. Vol. II: 1667–1670, Cambridge University Press, London-New York, 1968. Edited by D. T. Whiteside, with the assistance in publication of M. A. Hoskin. MR 0228320
D. G. Northcott, Finite free resolutions, Cambridge Tracts in Mathematics, No. 71, Cambridge University Press, Cambridge-New York-Melbourne, 1976. MR 0460383
N. Obatake, A. Shiu, X. Tang and A. Torres, Oscillations and bistability in a model of ERK regulation, arXiv:1903.02617[q-bio.MN].
A. Patel and S. Shvartsman, Outstanding questions in developmental ERK signaling, Development 145 (2018), dev143818.
Erwan Penchèvre, L’élimination en algèbre aux XVII$^{\mathrm e}$ et XVIII$^{\mathrm e}$ siècles, Historia Sci. (2) 14 (2004), no. 2, 101–117 (French, with French summary). MR 2145854
E. Penchèvre, Histoire de la théorie de l’élimination, Ph.D. Thesis, l’Université Paris VII, 2006.
E. Penchèvre, Etienne Bézout on elimination theory, 2016, arXiv:1606.03711 [math.HO].
E. Penchèvre, La théorie arithmétique des grandeurs algébriques de Kronecker (1882), 2018, arXiv:1801.04327 [math.HO].
Mercedes Pérez Millán and Alicia Dickenstein, The structure of MESSI biological systems, SIAM J. Appl. Dyn. Syst. 17 (2018), no. 2, 1650–1682. MR 3811780, DOI 10.1137/17M1113722
Mercedes Pérez Millán, Alicia Dickenstein, Anne Shiu, and Carsten Conradi, Chemical reaction systems with toric steady states, Bull. Math. Biol. 74 (2012), no. 5, 1027–1065. MR 2909119, DOI 10.1007/s11538-011-9685-x
M. Plecnik and R. Fearing, Finding only finite roots to large kinematic synthesis systems, J. Mech. Robotics 9 (2017), 021005.
S.-D. Poisson, Mémoire sur l’élimination dans les équations algébriques, J. Éc. polytech. Math. 4 (1802), 199–203. Reproduced in \cite[Appendix B]penchevrethesis.
H. Pollaczek-Geiringer, Über die Gliederung ebener Fachwerke, Z. Angew. Math. Mech. 7 (1927), 58–72.
M. Rojas, Review of \cite{gatermann3}, Mathematical Reviews, MR2178092 (2006m:13026).
Tien-Yien Li, J. Maurice Rojas, and Xiaoshen Wang, Counting real connected components of trinomial curve intersections and $m$-nomial hypersurfaces, Discrete Comput. Geom. 30 (2003), no. 3, 379–414. MR 2002964, DOI 10.1007/s00454-003-2834-8
A. Rosen, J. Sidman and L. Theran, Algebraic matroids in action, arXiv:1809.00865 [math.CO].
B. Roth, Rigid and flexible frameworks, Amer. Math. Monthly 88 (1981), no. 1, 6–21. MR 619413, DOI 10.2307/2320705
F. Rouillier, M.-F. Roy, and M. Safey El Din, Finding at least one point in each connected component of a real algebraic set defined by a single equation, J. Complexity 16 (2000), no. 4, 716–750. MR 1801591, DOI 10.1006/jcom.2000.0563
B. Rubinstein, H. Mattingly, A. Berezhkovskii and S. Shvartsman, Long-term dynamics of multisite phosphorylation, Mol. Biol. Cell. 27 (2017), 2331–2340.
AmirHosein Sadeghimanesh and Elisenda Feliu, Gröbner bases of reaction networks with intermediate species, Adv. in Appl. Math. 107 (2019), 74–101. MR 3921073, DOI 10.1016/j.aam.2019.02.006
AmirHosein Sadeghimanesh and Elisenda Feliu, The multistationarity structure of networks with intermediates and a binomial core network, Bull. Math. Biol. 81 (2019), no. 7, 2428–2462. MR 3977620, DOI 10.1007/s11538-019-00612-1
M. Sáez, C. Wiuf and E. Feliu, Nonnegative linear elimination for chemical reaction networks, 2018, arXiv:1807.00061[q-bio.MN].
M. Safey El Din, RAGlib library, https://www-polsys.lip6.fr/~safey/RAGLib/.
G. Salmon, Traité de Géométrie analytique a trois dimensiones, Paris, Gauthier-Villars, 1882.
Norbert Schappacher, A historical sketch of B. L. van der Waerden’s work in algebraic geometry: 1926–1946, Episodes in the history of modern algebra (1800–1950), Hist. Math., vol. 32, Amer. Math. Soc., Providence, RI, 2007, pp. 245–283. MR 2353499, DOI 10.1090/hmath/032/12
M. Schatzman, Numerical Analysis, Cambridge Univ. Press, Cambridge, 2002.
Hal Schenck, Alexandra Seceleanu, and Javid Validashti, Syzygies and singularities of tensor product surfaces of bidegree $(2,1)$, Math. Comp. 83 (2014), no. 287, 1337–1372. MR 3167461, DOI 10.1090/S0025-5718-2013-02764-0
I. J. Schoenberg, Remarks to Maurice Fréchet’s article “Sur la définition axiomatique d’une classe d’espace distanciés vectoriellement applicable sur l’espace de Hilbert” [MR1503246], Ann. of Math. (2) 36 (1935), no. 3, 724–732. MR 1503248, DOI 10.2307/1968654
Hermann Schubert, Kalkül der abzählenden Geometrie, Springer-Verlag, Berlin-New York, 1979 (German). Reprint of the 1879 original; With an introduction by Steven L. Kleiman. MR 555576
T. Sederberg, Computer Aided Geometric Design, 2016, tom.cs.byu.edu/~557/text/cagd.pdf.
T. Sederberg, D. Anderson and R. Goldman, Implicit representation of parametric curves and surfaces, Comput. Vision Graphics Image Process. 28 (1984), 72–84.
Tom Sederberg, Ron Goldman, and Hang Du, Implicitizing rational curves by the method of moving algebraic curves, J. Symbolic Comput. 23 (1997), no. 2-3, 153–175. Parametric algebraic curves and applications (Albuquerque, NM, 1995). MR 1448692, DOI 10.1006/jsco.1996.0081
T. Sederberg and F. Chen, Implicitization using moving curves and surfaces, in SIGGRAPH 95 Proceedings (R. Cook, ed.), ACM, New York, 1995, 301–308.
Thomas W. Sederberg, Takafumi Saito, Dong Xu Qi, and Krzysztof S. Klimaszewski, Curve implicitization using moving lines, Comput. Aided Geom. Design 11 (1994), no. 6, 687–706. MR 1305914, DOI 10.1016/0167-8396(94)90059-0
A. Seidenberg, A new decision method for elementary algebra, Ann. of Math. (2) 60 (1954), 365–374. MR 63994, DOI 10.2307/1969640
Jack Graver, Brigitte Servatius, and Herman Servatius, Combinatorial rigidity, Graduate Studies in Mathematics, vol. 2, American Mathematical Society, Providence, RI, 1993. MR 1251062, DOI 10.1090/gsm/002
F. Severi, Sul principio della conservazione del numero, Rend. Circ. Mat. Palermo 33 (1912), 313–327.
Li-Yong Shen and Ron Goldman, Strong $\mu$-bases for rational tensor product surfaces and extraneous factors associated to bad base points and anomalies at infinity, SIAM J. Appl. Algebra Geom. 1 (2017), no. 1, 328–351. MR 3672369, DOI 10.1137/16M1091952
G. Shinar and M. Feinberg, Structural sources of robustness in biochemical reaction networks, Science 327(5971) (2010), 1389–1391.
Anne Shiu and Timo de Wolff, Nondegenerate multistationarity in small reaction networks, Discrete Contin. Dyn. Syst. Ser. B 24 (2019), no. 6, 2683–2700. MR 3960599, DOI 10.3934/dcdsb.2018270
Jessica Sidman and Audrey St. John, The rigidity of frameworks: theory and applications, Notices Amer. Math. Soc. 64 (2017), no. 9, 973–978. MR 3699772, DOI 10.1090/noti1575
Steve Smale, Mathematical problems for the next century, Math. Intelligencer 20 (1998), no. 2, 7–15. MR 1631413, DOI 10.1007/BF03025291
Andrew J. Sommese and Charles W. Wampler II, The numerical solution of systems of polynomials, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2005. Arising in engineering and science. MR 2160078, DOI 10.1142/9789812567727
Andrew J. Sommese, Jan Verschelde, and Charles W. Wampler, Numerical irreducible decomposition using projections from points on the components, Symbolic computation: solving equations in algebra, geometry, and engineering (South Hadley, MA, 2000) Contemp. Math., vol. 286, Amer. Math. Soc., Providence, RI, 2001, pp. 37–51. MR 1874270, DOI 10.1090/conm/286/04753
Andrew J. Sommese, Jan Verschelde, and Charles W. Wampler, Symmetric functions applied to decomposing solution sets of polynomial systems, SIAM J. Numer. Anal. 40 (2002), no. 6, 2026–2046 (2003). MR 1974173, DOI 10.1137/S0036142901397101
Frank Sottile, Toric ideals, real toric varieties, and the moment map, Topics in algebraic geometry and geometric modeling, Contemp. Math., vol. 334, Amer. Math. Soc., Providence, RI, 2003, pp. 225–240. MR 2039975, DOI 10.1090/conm/334/05984
Giovanni Staglianò, A package for computations with classical resultants, J. Softw. Algebra Geom. 8 (2018), 21–30. MR 3820371, DOI 10.2140/jsag.2018.8.21
Hans J. Stetter, Numerical polynomial algebra, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2004. MR 2048781, DOI 10.1137/1.9780898717976
Audrey St. John, Generic rigidity of body-and-cad frameworks, Handbook of geometric constraint systems principles, Discrete Math. Appl. (Boca Raton), CRC Press, Boca Raton, FL, 2019, pp. 505–524. MR 3837068
Volker Strassen, Gaussian elimination is not optimal, Numer. Math. 13 (1969), 354–356. MR 248973, DOI 10.1007/BF02165411
Bernd Sturmfels, Algorithms in invariant theory, 2nd ed., Texts and Monographs in Symbolic Computation, SpringerWienNewYork, Vienna, 2008. MR 2667486
Bernd Sturmfels, Gröbner bases and convex polytopes, University Lecture Series, vol. 8, American Mathematical Society, Providence, RI, 1996. MR 1363949, DOI 10.1090/ulect/008
Bernd Sturmfels, Solving systems of polynomial equations, CBMS Regional Conference Series in Mathematics, vol. 97, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2002. MR 1925796, DOI 10.1090/cbms/097
Bernd Sturmfels, Introduction to resultants, Applications of computational algebraic geometry (San Diego, CA, 1997) Proc. Sympos. Appl. Math., vol. 53, Amer. Math. Soc., Providence, RI, 1998, pp. 25–39. MR 1602347, DOI 10.1090/psapm/053/1602347
B. Sturmfels, Can biology lead to new theorems?, Clay Mathematics Institute Annual Report 2005, Feature Article,13, 22–26.
B. Sturmfels and S. Sullivant, Toric ideals of phylogenetic invariants, J. Computational Biology 12 (2005), 204–228.
Bernd Sturmfels and Jenia Tevelev, Elimination theory for tropical varieties, Math. Res. Lett. 15 (2008), no. 3, 543–562. MR 2407231, DOI 10.4310/MRL.2008.v15.n3.a14
Bernd Sturmfels, Jenia Tevelev, and Josephine Yu, The Newton polytope of the implicit equation, Mosc. Math. J. 7 (2007), no. 2, 327–346, 351 (English, with English and Russian summaries). MR 2337885, DOI 10.17323/1609-4514-2007-7-2-327-346
J. Sylvester, A Method of determining by mere inspection the derivatives from two equations of any degree, Philos. Mag. XVI (1840), 132-135.
J. Sylvester, On a theory of the syzygetic relations of two rational integral functions, comprising an application to the theory of Sturm’s functions, and that of the greatest algebraical common measure, Philosophical Transactions 143 (1853), 407–548.
Tiong-Seng Tay, Rigidity of multigraphs. I. Linking rigid bodies in $n$-space, J. Combin. Theory Ser. B 36 (1984), no. 1, 95–112. MR 742389, DOI 10.1016/0095-8956(84)90016-9
Simon Telen and Marc Van Barel, A stabilized normal form algorithm for generic systems of polynomial equations, J. Comput. Appl. Math. 342 (2018), 119–132. MR 3808461, DOI 10.1016/j.cam.2018.04.021
S. Telen, B. Mourrain and M. Van Barel, Solving polynomial systems via a stabilized representation of quotient algebras, 2017, arXiv:1711.04543 [math.AG].
Matthew Thomson and Jeremy Gunawardena, The rational parameterisation theorem for multisite post-translational modification systems, J. Theoret. Biol. 261 (2009), no. 4, 626–636. MR 2974153, DOI 10.1016/j.jtbi.2009.09.003
M. Thomson and J. Gunawardena, Unlimited multistability in multisite phosphorylation systems, Nature 460(7252) (2009), 274–277.
Wolfgang Trinks, Über B. Buchbergers Verfahren, Systeme algebraischer Gleichungen zu lösen, J. Number Theory 10 (1978), no. 4, 475–488 (German, with English summary). MR 515056, DOI 10.1016/0022-314X(78)90019-7
E. Tschirnhaus, Methodus auferendi omnes terminos intermedios ex data equatione, Acta Eruditorium 2 (1683), 204–207. English Translation A method for removing all intermediate terms from a given equation by R. Green, ACM SIGSAM Bulletin 37, March 2003, 1–3.
Caroline Uhler, Geometry of maximum likelihood estimation in Gaussian graphical models, Ann. Statist. 40 (2012), no. 1, 238–261. MR 3014306, DOI 10.1214/11-AOS957
B. L. van der Waerden, Zur Nullstellentheorie der Polynomideale, Math. Ann. 96 (1927), no. 1, 183–208 (German). MR 1512314, DOI 10.1007/BF01209162
B. van der Waerden, Ein algebraisches Kriterium für die Lösbarkeit von homogenen Gleichungen, Nederl. Akad. Wetensch. Proc. 29 (1926), 142–149.
Bartel L. van der Waerden, Der Multiplizitätsbegriff der algebraischen Geometrie, Math. Ann. 97 (1927), no. 1, 756–774 (German). MR 1512387, DOI 10.1007/BF01447893
B. L. van der Waerden, The foundation of algebraic geometry from Severi to André Weil, Arch. History Exact Sci. 7 (1971), no. 3, 171–180. MR 1554142, DOI 10.1007/BF00357215
C. van Loan, Introduction to Scientific Computing: A Matrix-Vector Approach Using MATLAB, Second Edition, Pearson, Upper Saddle River, NJ, 2000.
Charles W. Wampler and Andrew J. Sommese, Numerical algebraic geometry and algebraic kinematics, Acta Numer. 20 (2011), 469–567. MR 2805156, DOI 10.1017/S0962492911000067
C. Wampler, A. Morgan and A. Sommese, Complete solution of the nine-point path synthesis problem for four-bar linkages, J. Mech. Des. 114 (1992), 153–159.
Liming Wang and Eduardo D. Sontag, On the number of steady states in a multiple futile cycle, J. Math. Biol. 57 (2008), no. 1, 29–52. MR 2393207, DOI 10.1007/s00285-007-0145-z
Edward Waring, Meditationes algebraicæ, American Mathematical Society, Providence, RI, 1991. Translated from the Latin, edited and with a foreword by Dennis Weeks; With an appendix by Franz X. Mayer, translated from the German by Weeks. MR 1146921
André Weil, Foundations of Algebraic Geometry, American Mathematical Society Colloquium Publications, Vol. 29, American Mathematical Society, New York, 1946. MR 0023093
Jerzy Weyman, Cohomology of vector bundles and syzygies, Cambridge Tracts in Mathematics, vol. 149, Cambridge University Press, Cambridge, 2003. MR 1988690, DOI 10.1017/CBO9780511546556
Neil White and Walter Whiteley, The algebraic geometry of motions of bar-and-body frameworks, SIAM J. Algebraic Discrete Methods 8 (1987), no. 1, 1–32. MR 872054, DOI 10.1137/0608001
Neil L. White and Walter Whiteley, The algebraic geometry of stresses in frameworks, SIAM J. Algebraic Discrete Methods 4 (1983), no. 4, 481–511. MR 721619, DOI 10.1137/0604049
Carsten Wiuf and Elisenda Feliu, Power-law kinetics and determinant criteria for the preclusion of multistationarity in networks of interacting species, SIAM J. Appl. Dyn. Syst. 12 (2013), no. 4, 1685–1721. MR 3116636, DOI 10.1137/120873388
Wen Tsün Wu, On the decision problem and the mechanization of theorem-proving in elementary geometry, Sci. Sinica 21 (1978), no. 2, 159–172. MR 479996
J. Verschelde, Algorithm 795: PHCpack: A general-purpose solver for polynomial systems by homotopy continuation, ACM Trans. Math. Software 25 (1999), 251–276.
Wolfram Research, Inc., Mathematica, Version 11.1, Champaign, IL, 2017.
Oscar Zariski, Generalized weight properties of the resultant of $n+1$ polynomials in $n$ indeterminates, Trans. Amer. Math. Soc. 41 (1937), no. 2, 249–265. MR 1501900, DOI 10.1090/S0002-9947-1937-1501900-8
O. Zariski and P. Samuel, Commutative Algebra, Volume II, Van Nostrand, Princeton, NJ, 1960. Reprint by Springer, New York, 1976.
Jianmin Zheng and Thomas W. Sederberg, A direct approach to computing the $\mu$-basis of planar rational curves, J. Symbolic Comput. 31 (2001), no. 5, 619–629. MR 1828707, DOI 10.1006/jsco.2001.0437
Günter M. Ziegler, Lectures on polytopes, Graduate Texts in Mathematics, vol. 152, Springer-Verlag, New York, 1995. MR 1311028, DOI 10.1007/978-1-4613-8431-1

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