A Unified Approach to Unimodality of Gaussian Polynomials

Christoph Koutschan; Ali Kemal Uncu; Elaine Wong

doi:10.1145/3597066.3597113

A Unified Approach to Unimodality of Gaussian Polynomials

Christoph Koutschan
, Ali Kemal Uncu
, Elaine Wong

Discrete Algorithms

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

Abstract

In 2013, Pak and Panova proved the strict unimodality property of q-binomial coefficients (as polynomials in q) based on the combinatorics of Young tableaux and the semigroup property of Kronecker coefficients. They showed it to be true for all ĝ.,", m ≥ 8 and a few other cases. We propose a different approach to this problem based on computer algebra, where we establish a closed form for the coefficients of these polynomials and then use cylindrical algebraic decomposition to identify exactly the range of coefficients where strict unimodality holds. This strategy allows us to tackle generalizations of the problem, e.g., to show unimodality with larger gaps or unimodality of related sequences. In particular, we present proofs of two additional cases of a conjecture by Stanley and Zanello.

Original language	English
Title of host publication	ISSAC 2023 - Proceedings of the 2023 International Symposium on Symbolic and Algebraic Computation
Editors	Gabriela Jeronimo
Publisher	Association for Computing Machinery
Pages	434-442
Number of pages	9
ISBN (Electronic)	9798400700392
DOIs	https://doi.org/10.1145/3597066.3597113
State	Published - Jul 24 2023
Event	48th International Symposium on Symbolic and Algebraic Computation, ISSAC 2023 - Tromso, Norway Duration: Jul 24 2023 → Jul 27 2023

Publication series

Name	ACM International Conference Proceeding Series

Conference

Conference	48th International Symposium on Symbolic and Algebraic Computation, ISSAC 2023
Country/Territory	Norway
City	Tromso
Period	07/24/23 → 07/27/23

Funding

C. Koutschan was supported by the Austrian Science Fund (FWF): I6130-N. A. K. Uncu was partially supported by the EPSRC grant EP/T015713/1 and partially by the Austrian Science Fund (FWF) P34501-N. E. Wong acknowledges that this manuscript has been partially authored by UT-Battelle, LLC under Contract No. DE-AC05-00OR22725 with the U.S. Department of Energy (DOE). The publisher acknowledges the US government license to provide public access under the DOE Public Access Plan C. Koutschan was supported by the Austrian Science Fund (FWF): I6130-N. A. K. Uncu was partially supported by the EPSRC grant EP/T015713/1 and partially by the Austrian Science Fund (FWF) P34501-N. E. Wong acknowledges that this manuscript has been partially authored by UT-Battelle, LLC under Contract No. DE-AC05-00OR22725 with the U.S. Department of Energy (DOE). The publisher acknowledges the US government license to provide public access under the DOE Public Access Plan.

Keywords

Gaussian polynomial
cylindrical algebraic decomposition
q-binomial coefficient
unimodality

Access to Document

10.1145/3597066.3597113

Cite this

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title = "A Unified Approach to Unimodality of Gaussian Polynomials",

abstract = "In 2013, Pak and Panova proved the strict unimodality property of q-binomial coefficients (as polynomials in q) based on the combinatorics of Young tableaux and the semigroup property of Kronecker coefficients. They showed it to be true for all ĝ.,{"}, m ≥ 8 and a few other cases. We propose a different approach to this problem based on computer algebra, where we establish a closed form for the coefficients of these polynomials and then use cylindrical algebraic decomposition to identify exactly the range of coefficients where strict unimodality holds. This strategy allows us to tackle generalizations of the problem, e.g., to show unimodality with larger gaps or unimodality of related sequences. In particular, we present proofs of two additional cases of a conjecture by Stanley and Zanello.",

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Koutschan, C, Uncu, AK & Wong, E 2023, A Unified Approach to Unimodality of Gaussian Polynomials. in G Jeronimo (ed.), ISSAC 2023 - Proceedings of the 2023 International Symposium on Symbolic and Algebraic Computation. ACM International Conference Proceeding Series, Association for Computing Machinery, pp. 434-442, 48th International Symposium on Symbolic and Algebraic Computation, ISSAC 2023, Tromso, Norway, 07/24/23. https://doi.org/10.1145/3597066.3597113

A Unified Approach to Unimodality of Gaussian Polynomials. / Koutschan, Christoph; Uncu, Ali Kemal; Wong, Elaine.
ISSAC 2023 - Proceedings of the 2023 International Symposium on Symbolic and Algebraic Computation. ed. / Gabriela Jeronimo. Association for Computing Machinery, 2023. p. 434-442 (ACM International Conference Proceeding Series).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

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N2 - In 2013, Pak and Panova proved the strict unimodality property of q-binomial coefficients (as polynomials in q) based on the combinatorics of Young tableaux and the semigroup property of Kronecker coefficients. They showed it to be true for all ĝ.,", m ≥ 8 and a few other cases. We propose a different approach to this problem based on computer algebra, where we establish a closed form for the coefficients of these polynomials and then use cylindrical algebraic decomposition to identify exactly the range of coefficients where strict unimodality holds. This strategy allows us to tackle generalizations of the problem, e.g., to show unimodality with larger gaps or unimodality of related sequences. In particular, we present proofs of two additional cases of a conjecture by Stanley and Zanello.

AB - In 2013, Pak and Panova proved the strict unimodality property of q-binomial coefficients (as polynomials in q) based on the combinatorics of Young tableaux and the semigroup property of Kronecker coefficients. They showed it to be true for all ĝ.,", m ≥ 8 and a few other cases. We propose a different approach to this problem based on computer algebra, where we establish a closed form for the coefficients of these polynomials and then use cylindrical algebraic decomposition to identify exactly the range of coefficients where strict unimodality holds. This strategy allows us to tackle generalizations of the problem, e.g., to show unimodality with larger gaps or unimodality of related sequences. In particular, we present proofs of two additional cases of a conjecture by Stanley and Zanello.

KW - Gaussian polynomial

KW - cylindrical algebraic decomposition

KW - q-binomial coefficient

KW - unimodality

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ER -

A Unified Approach to Unimodality of Gaussian Polynomials

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