The multiplicity sequence of monomial ideals

Sudipta Das, Jonathan Montaño, Aniketh Sivakumar

arXiv:2608.12731v1 Primary: math.AC Secondary: math.CO

Abstract

We give a convex-geometric formula for the multiplicity sequence of a monomial ideal in terms of mixed volumes of polytopes constructed from its Newton polyhedron. We also construct a counterexample to a conjecture of Achilles and Manaresi proposing a different volume formula for the multiplicity sequence. Finally, we derive a mixed-volume formula for the mixed multiplicities of arbitrary monomial ideals.

Disclosure

“Acknowledgments The second author is grateful to Federico Castillo for helpful discussions about mixed volumes of polytopes. The second author was partially funded by NSF Grant DMS #2401522. ChatGPT and Claude were used to find relevant references and to improve the readability of some sentences throughout the article. References [1] R. Achilles and M. Manaresi. Multiplicities of a”
PDF page 14

Classification: Rewriting existing author-written text
Multiplier: 4
Verified: 2026-08-23T05:12:27.368241Z

Structural counts

Pages 15 pdf

Theorems 4 source

Lemmas 7 source

Propositions 4 source

Corollaries 2 source

Definitions 3 source

Displayed equations 71 source

Bibliography entries 28 source

Appendix pages 0 estimated

Count notes

Source counts use the expanded primary TeX file Clean_monomial_multSeq.tex.
Appendix pages include the first PDF page with an explicit Appendix heading through the final page.