Real polynomials with given multiplicities of real roots: Complete conjectural description of homology

Abstract

Following \cite {KSW} we continue the study the cellular complexes formed by polynomials of a given degree having a given sequence of multiplicities of real roots. A computer-assisted calculation disproves the earlier conjecture that homology of one point compactification of the closure of such cell is concentrated in at most one degree. Namely, for $ω=(3,1,1,3)$ in degree $d=18$, the reduced homology is $\ZZ^2$ in degree $7$. The obstruction is already visible in a signed cell count, whose value is $-2$. We relate that count to the rational signed weight enumerator $F_ω(t)=\sum_{η\preceqω}(-1)^{\elln(η)}t^{|η|}$. For the counterexample, $F_ω(t)=-t^{14}/(1+t^2)^2$, which gives an exact linear formula for the Euler characteristic and forces the total rational Betti number to be unbounded. We prove a number of results and formulate a complete conjecture describing the above homology.

Disclosure

“ith |ω| ≤ 11, d ≤ 16. These checks establish the finite computations stated as theorems or computer-assisted results. AI-statement The main counterexample in Theorem 1 was discovered by Claude Opus 5 and the author formulated the main conjecture Conjecture 5 of this note which he then check using Claude Opus 5 and GPT 5.6 Sol. Although most of the claims here remain conjectural they provide a complete description of homology of”

PDF page 10

Classification

Substantial mathematical content or result generation

Multiplier

10

Verified

2026-08-22T18:13:57.574788Z

Structural counts