Euler characteristic of the space of real multivariate irreducible polynomials

Hyde, Trevor

doi:10.1090/proc/15849

Euler characteristic of the space of real multivariate irreducible polynomials
HTML articles powered by AMS MathViewer

by Trevor Hyde PDF

Proc. Amer. Math. Soc. 150 (2022), 2331-2343 Request permission

Abstract:

We compute the compactly supported Euler characteristic of the space of degree $d$ irreducible polynomials in $n$ variables with real coefficients and show that the values are given by the digits in the so-called balanced binary expansion of the number of variables $n$.

References

Raoul Bott and Loring W. Tu, Differential forms in algebraic topology, Graduate Texts in Mathematics, vol. 82, Springer-Verlag, New York-Berlin, 1982. MR 658304, DOI 10.1007/978-1-4757-3951-0
Weiyan Chen, Stability of the cohomology of the space of complex irreducible polynomials in several variables, Int. Math. Res. Not. IMRN 22 (2021), 17256–17276. MR 4345827, DOI 10.1093/imrn/rnz296
Jesse Elliott, Binomial rings, integer-valued polynomials, and $\lambda$-rings, J. Pure Appl. Algebra 207 (2006), no. 1, 165–185. MR 2244389, DOI 10.1016/j.jpaa.2005.09.003
N. J. Fine, Binomial coefficients modulo a prime, Amer. Math. Monthly 54 (1947), 589–592. MR 23257, DOI 10.2307/2304500
Philip Hall, The Edmonton notes on nilpotent groups, Queen Mary College Mathematics Notes, Queen Mary College, Mathematics Department, London, 1969. MR 0283083
Allen Hatcher, Algebraic topology, Cambridge University Press, Cambridge, 2002. MR 1867354
Trevor Hyde, Liminal reciprocity and factorization statistics, Algebr. Comb. 2 (2019), no. 4, 521–539. MR 3997509, DOI 10.5802/alco.34
T. Hyde, Cyclotomic factors of necklace polynomials, preprint, arXiv:1811.08601, 2021.
Trevor Hyde, Polynomial Statistics, Necklace Polynomials, and the Arithmetic Dynamical Mordell-Lang Conjecture, ProQuest LLC, Ann Arbor, MI, 2019. Thesis (Ph.D.)–University of Michigan. MR 4071618
Edouard Lucas, Theorie des Fonctions Numeriques Simplement Periodiques, Amer. J. Math. 1 (1878), no. 4, 289–321 (French). MR 1505176, DOI 10.2307/2369373
I. G. MacDonald, The Poincaré polynomial of a symmetric product, Math. Proc. Camb. Philos. Soc., 58, no. 4, (1962), DOI: 10.1017/S0305004100040573.
Edwin H. Spanier, Algebraic topology, Springer-Verlag, New York-Berlin, 1981. Corrected reprint. MR 666554, DOI 10.1007/978-1-4684-9322-1
R. Vakil, Arizona winter school notes, available at http://swc.math.arizona.edu/aws/2015/2015VakilNotes.pdf.