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arXiv · 1402.4000

An exact degree for multivariate special polynomials

Rudolph Bronson Perkins
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Abstract

We introduce certain special polynomials in an arbitrary number of indeterminates over a finite field. These polynomials generalize the special polynomials associated to the Goss zeta function and Goss-Dirichlet $L$-functions over the ring of polynomials in one indeterminate over a finite field and also capture the special values at non-positive integers of $L$-series associated to Drinfeld modules over Tate algebras defined over the same ring. We compute the exact degree in $t_0$ of these special polynomials and show that this degree is an invariant for a natural action of Goss' group of digit permutations. Finally, we characterize the vanishing of these multivariate special polynomials at $t_0=1$. This gives rise to a notion of trivial zeros for our polynomials generalizing that of the Goss zeta function mentioned above.

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Rudolph Bronson Perkins. 2014-02-17. An exact degree for multivariate special polynomials. https://doi.org/10.1016/j.jnt.2014.02.022

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