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PreprintCC BY 4.0arXiv

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

  • Boris Shapiro
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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 ZZ2 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 Fω(t)=∑η⪯ω(−1)elln(η)t|η|. For the counterexample, Fω(t)=−t14/(1+t2)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.

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Cite

BibTeXRIS
BibTeX
@misc{shapiro2026real,
  title = {{Real polynomials with given multiplicities of real roots: Complete conjectural description of homology}},
  author = {Boris Shapiro},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2608.19733},
  eprint = {2608.19733},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.19733},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Real polynomials with given multiplicities of real roots: Complete conjectural description of homology
AU  - Boris Shapiro
PY  - 2026
DA  - 2026/09/06
PB  - arXiv
DO  - 10.48550/arxiv.2608.19733
UR  - https://arxiv.org/abs/2608.19733
AB  - 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)=∑_η⪯ω(−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.
ER  -