Reproduce “Symmetric Functions over Finite Fields” on OpenResearch
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About this paper
Mihai Prunescu
Mihai Prunescu developed an algorithm to compute the dimension and construct a vector space basis for the ideal of algebraic relations among elementary symmetric functions over finite fields. The work establishes this dimension as $q^n - \binom{n+q-1}{n}$ and proves that the basis polynomials generated by the algorithm have coefficients in the prime field F$_p$.
Key insights
- The dimension of the ideal of algebraic relations among elementary symmetric functions over F$_q$ is precisely $q^n - \binom{n+q-1}{n}$.
- The vector space basis for these algebraic relations, as generated by the proposed algorithm, consists of polynomials whose coefficients are exclusively in the prime field F$_p$ of F$_q$.
- Asymptotic analysis shows that for a fixed field size $q$, almost all formal polynomials in elementary symmetric functions become relations as the number of variables $n$ grows.
Summary by alphaXiv.
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