Stability of the Cohomology of the Space of Complex Irreducible Polynomials in Several Variables

Chen, Weiyan

doi:10.1093/imrn/rnz296

Cited by 2 publications

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“…Chen [1] proved that the singular and compactly supported cohomology of the spaces Irr d,n (C) stabilizes in certain cohomological degree regimes as n → ∞. We do not know whether Chen's methods can be adapted to analyze the compactly supported cohomology of Irr d,n (R).…”

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Euler characteristic of the space of real multivariate irreducible polynomials

Hyde¹

2020

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Given a field K and integers d, n ≥ 1, let Poly d,n (K) denote the set of all monic total degree d polynomials in n variables with coefficients in K (see Definition 2.1.) Let Irr d,n (K) ⊆ Poly d,n (K) denote the subset of all polynomials which are irreducible over K. Our main result expresses the compactly supported Euler characteristic of Irr d,n (R) in terms of the so-called balanced binary expansion of the number of variables n. The balanced binary expansion of an integer n ≥ 1 is the unique expressionwhere 0 ≤ k 0 < k 1 < . . . < k 2m is a strictly increasing sequence of natural numbers of even length and the signs on the right hand side alternate. Theorem 1.1. Let n ≥ 1 be an integer with balanced binary expansion n = ℓ k=0 b k 2 k and let χ c denote the compactly supported Euler characteristic. Then χ c (Irr d,n (R)) = b k if d = 2 k , 0 otherwise. Example 1.2. The balanced binary expansion of n = 13 is 13 = 2 4 − 2 2 + 2 1 − 1. Thus Theorem 1.1 implies that χ c (Irr d,13 (R)) =      1 if d = 2 or 16, −1 if d = 1 or 4, 0 otherwise.

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confidence: 99%