Exact counterexample family

Determine the reduced homology of the one-point compactifications of the closures of the strata generated by the composition (3,1,1,3) for every even degree d≥14, proving that it is concentrated in degree d/2−2, has rank floor((d−10)/4), and vanishes in every other degree.

Background

The paper gives a computer-assisted calculation showing that for degree d=18 the stratum associated with (3,1,1,3) has reduced homology of rank two in degree seven. Its signed Euler characteristic is −2, and an exact rational weight enumerator implies that the absolute Euler characteristic grows linearly with d.

The conjecture extends this computation to every even d≥14 and asserts that the Euler-characteristic lower bound is the complete homological answer. It was directly verified only through d=22, so the global concentration and rank formula remain unresolved.

References

What remains conjectural is the stronger assertion that the homology is always concentrated in the observed degree and has rank exactly the absolute Euler characteristic.

— Real polynomials with given multiplicities of real roots: Complete conjectural description of homology  (2608.19733 - Shapiro, 20 Aug 2026) in Conjecture 3.1, Section 3

The weaker statement needed merely to decide vanishing is the following.

— Real polynomials with given multiplicities of real roots: Complete conjectural description of homology  (2608.19733 - Shapiro, 20 Aug 2026) in Conjecture 6.1, Section 6.1