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 $\ZZ<sup>2$ 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 $F_ω(t)=\sum_{η\preceqω}(-1)<sup>{\elln(η)}t<sup>{|η|}$. For the counterexample, $F_ω(t)=-t<sup>{14}/(1+t<sup>2)<sup>2$, 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.

• The paper constructs a finite cellular complex for each root-multiplicity composition and uses computer-assisted integral reductions to show that the rank-one conjecture fails, with $(3,1,1,3)$ at degree 18 yielding homology $\mathbb{Z}^2$ in degree 7.
• It proves that the Euler-characteristic generating function is rational and shows that $|\chi_d(3,1,1,3)|=\lfloor(d-14)/4\rfloor+1$ for even $d\ge14$, establishing unbounded total rational Betti numbers.
• It proposes a comprehensive conjectural framework involving freeness, homology concentration, anchored compositions, structured denominators, and degree laws while identifying discrete Morse theory and avoidance complexes as promising proof tools.

Overview

This paper studies the topology of strata in the space of monic real univariate polynomials of fixed degree, where the strata are indexed by compositions recording the multiplicities of real roots from left to right. Building on the cellular model of Katz–Shapiro–Welker (Katz et al., 2021), the author associates to each composition $\omega$ and admissible degree $d$ a finite reduced cellular complex $C_*(\omega;d)$ computing the reduced homology of the one-point compactification $\mathcal{P}_d^{\langle\omega\rangle}$ of the closed principal stratum. The paper's contributions are threefold: a computer-assisted disproof of two earlier conjectures (rank-one homology and local absorption), an exact rational generating function framework for Euler characteristics that yields unbounded Betti numbers, and a comprehensive set of conjectures that together would give a complete description of when this homology is nonzero, in which degree it lives, and what its rank is.

The cellular model and a membership criterion

For a composition $\eta$, the paper uses $|\eta|$ for its norm, $\ell(\eta)$ for its length, and $|\eta|' = |\eta| - \ell(\eta)$. The poset $\langle\omega\rangle$ consists of descendants obtained by merging adjacent parts (modeling collisions of real roots) and inserting parts equal to $2$ (modeling conjugate pairs becoming real). For admissible pairs ($d$0, $d$1), cells are descendants with $d$2, graded by $d$3, with a differential combining signed merges and insertions.

A key technical tool is the membership criterion (Lemma 2.1): $d$4 if and only if the parts of $d$5 can be partitioned into consecutive blocks assigned to parts of $d$6, such that each $d$7 dominates the corresponding block sum in magnitude and parity. This replaces closure under iterated moves by a finite-state test: after reading a prefix of $d$8, one tracks the set of possible numbers of $d$9-parts consumed. The paper notes that carrying the full subset is essential—a greedy assignment fails already for $C_*(\omega;d)$0, $C_*(\omega;d)$1. Geometrically, the closed stratum is identified with polynomials $C_*(\omega;d)$2 with $C_*(\omega;d)$3 and $C_*(\omega;d)$4 monic nonnegative on $C_*(\omega;d)$5, so the reduced homology coincides with Borel–Moore homology of this semialgebraic set.

Disproof of the rank-one conjecture

The central result refutes the earlier conjecture that reduced homology is either zero or free of rank one:

Main counterexample: for $C_*(\omega;d)$6 and $C_*(\omega;d)$7, the reduced homology is $C_*(\omega;d)$8 concentrated in degree $C_*(\omega;d)$9. The complex has $\mathcal{P}_d^{\langle\omega\rangle}$0 cells; both alternating sums (by length and by reduced norm) equal $\mathcal{P}_d^{\langle\omega\rangle}$1, so the rank jump is visible at the level of signed cell counts before any boundary matrix is constructed. Verification was thorough: cell generation by move-closure and by the membership criterion agree; integral elementary cancellations reduce the complex to two generators in degree 7 (an exact chain equivalence over $\mathcal{P}_d^{\langle\omega\rangle}$2, ruling out torsion); and unreduced elimination over $\mathcal{P}_d^{\langle\omega\rangle}$3 for six primes gives identical Betti numbers.

A second casualty is the "local absorption" guess that $\mathcal{P}_d^{\langle\omega\rangle}$4 always preserves homology. For $\mathcal{P}_d^{\langle\omega\rangle}$5, $\mathcal{P}_d^{\langle\omega\rangle}$6, the stratum for $\mathcal{P}_d^{\langle\omega\rangle}$7 has $\mathcal{P}_d^{\langle\omega\rangle}$8 in degree 4 while the stratum for $\mathcal{P}_d^{\langle\omega\rangle}$9 is acyclic. Finite searches show no counterexample with $\eta$0 and $\eta$1, and the computed failure pattern for the third move (nonminimal odd cases all in one congruence class mod $\eta$2) motivates a precise conditional absorption law, still conjectural.

Rational enumerators and unbounded Betti numbers

The grading collapses at the Euler-characteristic level: since every descendant shares the parity of $\eta$3, the sign $\eta$4 equals $\eta$5, giving

$\eta$6

where $\eta$7. A transfer-matrix argument using the subset automaton proves $\eta$8 is rational for every $\eta$9: once a letter weight reaches $|\eta|$0, transitions depend only on parity, so transition-matrix entries lie in a finitely generated module over $|\eta|$1.

For the counterexample composition, exact reduction gives $|\eta|$2, whence $|\eta|$3 for even $|\eta|$4. Since total rational Betti number dominates $|\eta|$5, the total rational Betti number of these strata grows without bound as $|\eta|$6 varies—an unconditional consequence independent of any concentration hypothesis. The stronger claim that the homology itself has rank exactly $|\eta|$7 in degree $|\eta|$8 is verified only through $|\eta|$9.

Repeated copies of the pattern amplify the effect: for $\ell(\eta)$0 consisting of $\ell(\eta)$1 threes separated by pairs of ones, the eventual quasi-polynomial degrees of $\ell(\eta)$2 are $\ell(\eta)$3 for $\ell(\eta)$4, with values reaching $\ell(\eta)$5, $\ell(\eta)$6, and $\ell(\eta)$7 respectively. Whether pole orders are unbounded in $\ell(\eta)$8 is conjectural.

The revised conjectural picture

The bulk of the paper formulates interlocking conjectures supported by extensive computation (all compositions of norm $\ell(\eta)$9 against admissible degrees through $|\eta|' = |\eta| - \ell(\eta)$0; thousands of pairs cross-checked):

• Concentration and freeness: reduced homology is always free and concentrated in at most one degree. No torsion or multi-degree homology appeared in any computed range.
• Euler characteristic detects vanishing: $|\eta|' = |\eta| - \ell(\eta)$1 iff the reduced homology vanishes. Only the forward implication is conjectural; combined with rationality it would reduce vanishing to extracting one coefficient of a rational function.
• Combinatorial support: $|\eta|' = |\eta| - \ell(\eta)$2 iff $|\eta|' = |\eta| - \ell(\eta)$3 is not "anchored" (i.e., unless $|\eta|' = |\eta| - \ell(\eta)$4 or both extreme parts are odd). There is a geometric heuristic—an even leading part $|\eta|' = |\eta| - \ell(\eta)$5 is nonnegative and absorbable into the free factor $|\eta|' = |\eta| - \ell(\eta)$6, killing Borel–Moore homology—but no proof.
• Block-factor denominator: for anchored $|\eta|' = |\eta| - \ell(\eta)$7 with intervening block weights $|\eta|' = |\eta| - \ell(\eta)$8, the reduced denominator of $|\eta|' = |\eta| - \ell(\eta)$9 should be a product of factors $\langle\omega\rangle$0, making the support of $\langle\omega\rangle$1 eventually a finite union of arithmetic progressions. This held for $\langle\omega\rangle$2 of $\langle\omega\rangle$3 tested anchored compositions, but the selection rule is unknown and provably not local: $\langle\omega\rangle$4 has block sequence $\langle\omega\rangle$5 yet only one $\langle\omega\rangle$6 factor.
• Degree laws: at the first degree $\langle\omega\rangle$7 with nonzero homology, concentration occurs in degree $\langle\omega\rangle$8 (number of odd parts plus one); in the generic single-period class, the degree follows the affine law $\langle\omega\rangle$9 along $2$0. This held for all $2$1 single-period compositions in the test range. Crucially, the unrestricted formula fails in multi-period cases: $2$2 shows slope $2$3 rather than $2$4, and the correct replacement is open.
• One-part resonance: for $2$5, homology is $2$6 in degree 1 at $2$7 if $2$8 is odd, and $2$9 in degree $d$00 exactly when $d$01 if $d$02 is even (verified for $d$03, $d$04).

Two structural tools are developed toward proofs: an avoidance-complex short exact sequence yielding suspension isomorphisms $d$05 for $d$06, suggesting a discrete Morse matching on pattern-avoidance as a route to concentration; and an all-even reduction halving weights when $d$07 and $d$08 are both even.

Limitations and open questions

The paper is candid that most of its claims rest on computation within bounded ranges. The main theorem is computer-assisted; the complete conjectural description—concentration, freeness, Euler-detects-vanishing, support, denominators, and degree laws—is unproven. Specific gaps include: no proof of the forward implication in the vanishing criterion (torsion-only homology or inter-degree cancellation remains possible); no sign-reversing involution establishing the anchored-support criterion; no discrete Morse matching on the avoidance complex; no rule selecting denominator factors among block weights; no resolution of the multi-period degree law beyond the $d$09 starting point; and no structural explanation of the rank jump across $d$10 at $d$11. The absorption conjecture's congruence condition for the third move is likewise unproven, though consistent with all data through $d$12.

Conclusion

The paper converts a failed rank-one conjecture into a sharper and more informative theory. Its unconditional contributions—the rationality of $d$13, the linear-growth formula for $d$14, and hence unboundedness of total rational Betti numbers—are solid, with the counterexample verified integrally and modulo several primes. Its conjectural framework, if confirmed, would constitute a complete answer to when, where, and how large the homology of these polynomial strata is, reducing the questions to coefficient extraction from explicitly structured rational functions. The main open task is to replace the computational evidence with proofs, with the avoidance complex and discrete Morse theory identified as the most promising avenues.