Inequalities among Symmetric Polynomial Functions: Counter-examples and New Conjectures
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| Format: | Preprint |
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2023
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| _version_ | 1866915284295614464 |
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| author | Xu, Jia Yao, Yong |
| author_facet | Xu, Jia Yao, Yong |
| contents | Inequalities among symmetric polynomial functions are fundamental questions in mathematics and have various applications in science and engineering. This paper investigates a beautiful and inspiring conjecture, proposed by Cuttler, Greene and Skandera in 2011, on inequalities among the complete homogeneous symmetric polynomial function $H_{n,λ}$: It states that the inequality $H_{n,λ}\leq H_{n,μ}$ implies majorization order $λ\preceqμ$. The conjecture is a close analogy with other known results on Muirhead-type inequalities. In 2021, Heaton and Shankar disproved the conjecture by showing a counterexample for number of variables $n=3$ and degree $d=8$. They then asked whether the conjecture is true when $n$ is sufficiently large. In this paper, we show, by a family of counter-examples, that the conjecture does not hold for any $n$ and any $d$ as long as $n\geq2$ and $d\geq8$. Based on the insights gained from the counter-examples, we propose a new conjecture for the inequality $H_{n,λ}\leq H_{n,μ}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2305_19830 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Inequalities among Symmetric Polynomial Functions: Counter-examples and New Conjectures Xu, Jia Yao, Yong Combinatorics Symbolic Computation 05E05 14P99 90C22 Inequalities among symmetric polynomial functions are fundamental questions in mathematics and have various applications in science and engineering. This paper investigates a beautiful and inspiring conjecture, proposed by Cuttler, Greene and Skandera in 2011, on inequalities among the complete homogeneous symmetric polynomial function $H_{n,λ}$: It states that the inequality $H_{n,λ}\leq H_{n,μ}$ implies majorization order $λ\preceqμ$. The conjecture is a close analogy with other known results on Muirhead-type inequalities. In 2021, Heaton and Shankar disproved the conjecture by showing a counterexample for number of variables $n=3$ and degree $d=8$. They then asked whether the conjecture is true when $n$ is sufficiently large. In this paper, we show, by a family of counter-examples, that the conjecture does not hold for any $n$ and any $d$ as long as $n\geq2$ and $d\geq8$. Based on the insights gained from the counter-examples, we propose a new conjecture for the inequality $H_{n,λ}\leq H_{n,μ}$. |
| title | Inequalities among Symmetric Polynomial Functions: Counter-examples and New Conjectures |
| topic | Combinatorics Symbolic Computation 05E05 14P99 90C22 |
| url | https://arxiv.org/abs/2305.19830 |