The BKK root count in 𝐂ⁿ

Li, T.; Wang, Xiaoshen

doi:10.1090/S0025-5718-96-00778-8

The BKK root count in $\mathbf {C}^n$
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by T. Y. Li and Xiaoshen Wang PDF

Math. Comp. 65 (1996), 1477-1484 Request permission

Abstract:

The root count developed by Bernshtein, Kushnirenko and Khovanskii only counts the number of isolated zeros of a polynomial system in the algebraic torus $(\mathbf {C}^*)^n$. In this paper, we modify this bound slightly so that it counts the number of isolated zeros in $\mathbf {C}^n$. Our bound is, apparently, significantly sharper than the recent root counts found by Rojas and in many cases easier to compute. As a consequence of our result, the Huber-Sturmfels homotopy for finding all the isolated zeros of a polynomial system in $(\mathbf {C}^*)^n$ can be slightly modified to obtain all the isolated zeros in $\mathbf {C}^n$.

References

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
Felix E. Browder, Nonlinear mappings of analytic type in Banach spaces, Math. Ann. 185 (1970), 259–278. MR 259683, DOI 10.1007/BF01349949
T. Bonnesen and W. Fenchel, Theory of convex bodies, BCS Associates, Moscow, ID, 1987. Translated from the German and edited by L. Boron, C. Christenson and B. Smith. MR 920366
Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. MR 0463157
Birkett Huber and Bernd Sturmfels, A polyhedral method for solving sparse polynomial systems, Math. Comp. 64 (1995), no. 212, 1541–1555. MR 1297471, DOI 10.1090/S0025-5718-1995-1297471-4
A. G. Hovanskiĭ, Newton polyhedra, and the genus of complete intersections, Funktsional. Anal. i Prilozhen. 12 (1978), no. 1, 51–61 (Russian). MR 487230
A. G. Kušnirenko, Newton polyhedra and Bezout’s theorem, Funkcional. Anal. i Priložen. 10 (1976), no. 3, 82–83. (Russian). MR 0422272
T.-Y. Li, Tim Sauer, and James A. Yorke, The random product homotopy and deficient polynomial systems, Numer. Math. 51 (1987), no. 5, 481–500. MR 910860, DOI 10.1007/BF01400351
J. Maurice Rojas, A convex geometric approach to counting the roots of a polynomial system, Theoret. Comput. Sci. 133 (1994), no. 1, 105–140. Selected papers of the Workshop on Continuous Algorithms and Complexity (Barcelona, 1993). MR 1294429, DOI 10.1016/0304-3975(93)00062-A
J. M. Rojas and X. Wang, Counting affine roots via pointed Newton polytopes, to appear: J. of Complexity.
I. R. Shafarevich, Basic algebraic geometry, Springer Study Edition, Springer-Verlag, Berlin-New York, 1977. Translated from the Russian by K. A. Hirsch; Revised printing of Grundlehren der mathematischen Wissenschaften, Vol. 213, 1974. MR 0447223
Jan Verschelde and Ronald Cools, Symbolic homotopy construction, Appl. Algebra Engrg. Comm. Comput. 4 (1993), no. 3, 169–183. MR 1226323, DOI 10.1007/BF01202036
Jan Verschelde, Pierre Verlinden, and Ronald Cools, Homotopies exploiting Newton polytopes for solving sparse polynomial systems, SIAM J. Numer. Anal. 31 (1994), no. 3, 915–930. MR 1275120, DOI 10.1137/0731049