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Monotonicity of Turán-ratios of symmetric polynomials

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Contributed by Suvrit Sra ↗

Version 1 / Nov 02, 2026 / CC BY 4.0

Abstract

We study the Turán-ratio Sλ,i(x)2/(Sλ,i−1(x)Sλ,i+1(x))S_{\lambda,i}(x)^2/(S_{\lambda,i-1}(x)S_{\lambda,i+1}(x)), where Sλ,i(x):=sλ(1[n−i],x[i])S_{\lambda,i}(x):=s_\lambda(1^{[n-i]},x^{[i]}), and sλs_\lambda is the Schur polynomial in nn variables (indexed by partition λ\lambda) with ii arguments equal to xx and n−in-i of them equal to 11. Specifically, we prove that this Turán-ratio is strictly increasing for x>1x>1 and 1≤i<n1\le i<n (unless all nn padded parts of λ\lambda are equal, then it is identically one). For one-row shapes, our result answers a question of Ait-Haddou about complete homogeneous symmetric polynomials. For this case, we also offer a second proof through the zeros of Meixner polynomials that gives an explicit positive formula for the logarithmic derivative. We conclude with a Turán-ratio conjecture for Macdonald polynomials.

Provenance statement

The results in this paper were developed through extensive use of an AI assistant (OpenAI's GPT Pro). None of the claims were obtained zero-shot: they emerged through sustained back-and-forth discussion, including failed attempts, counterexamples, audits, and repairs.

Tools used

OpenAI
GPT Pro

References

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  3. L. A. Gonz á lez-Serrano and E. A. Maximenko. Bialternant formula for Schur polynomials with repeating variables. Linear and Multilinear Algebra , 2025. 10.1080/03081087.2025.2464639. https://arxiv.org/abs/2312.15680.DOI
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Version history

  1. v1Submitted by Suvrit SraInitial depositCurrentOct 31, 2026