Initial degree rule

Prove that for every anchored composition ω that is not a one-part composition of odd size, the first admissible degree d₀(ω) with nonzero principal homology has that homology concentrated in degree m(ω)+1, and establish the corresponding degree-one assertion for odd one-part compositions.

Background

The paper studies not only whether homology is nonzero but also the homological degree in which it occurs. Computations support a robust rule for the first degree at which nonzero homology appears, based on the number of odd parts of the composition.

The rule was supported by extensive finite computations, but it is presented as a conjecture rather than proved generally.

References

The data support an initial-degree rule more robustly than a universal slope rule.

— Real polynomials with given multiplicities of real roots: Complete conjectural description of homology  (2608.19733 - Shapiro, 20 Aug 2026) in Conjecture 6.5, Section 6.4

The one-part case has a particularly simple observed answer.

— Real polynomials with given multiplicities of real roots: Complete conjectural description of homology  (2608.19733 - Shapiro, 20 Aug 2026) in Conjecture 6.7, Section 6.4