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Given a finite abelian group G, a finite set D, and a mapping \(f:D\rightarrow G\) , we find the number of r-subsets \(S\subseteq D\) where for \(b\in G\) , \(\begin{aligned} \sum _{x\in S}f(x)=b. \end{aligned}\) We count degree n monic polynomials over \({\mathbb {F}}_q\) with r distinct roots in a set \(D\subseteq {\mathbb {F}}_q\) when the leading terms of degree at least \(n-\ell \) are fixed. We obtain new formulas for \(\ell =2\) when D is an arbitrary subfield of \({\mathbb {F}}_q\) with q odd.

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Counting Polynomials with Distinct Roots Using Subset Sum

  • 会议论文

  • 发表日期：

    2025年

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  • Simon Kuttner,
  • Zhicheng Gao,
  • Qiang Wang

摘要

Given a finite abelian group G, a finite set D, and a mapping \(f:D\rightarrow G\) , we find the number of r-subsets \(S\subseteq D\) where for \(b\in G\) , \(\begin{aligned} \sum _{x\in S}f(x)=b. \end{aligned}\) We count degree n monic polynomials over \({\mathbb {F}}_q\) with r distinct roots in a set \(D\subseteq {\mathbb {F}}_q\) when the leading terms of degree at least \(n-\ell \) are fixed. We obtain new formulas for \(\ell =2\) when D is an arbitrary subfield of \({\mathbb {F}}_q\) with q odd.