On the characterization of minimal value set polynomials

Details On the characterization of minimal value set polynomials On the characterization of minimal value set polynomials

2011-08-09

arxiv.org/abs/1108.1852v1

Mathematics

Number Theory

19 pages

Scientific paper

We give an explicit characterization of all minimal value set polynomials in $\F_q[x]$ whose set of values is a subfield $\F_{q'}$ of $\F_{q}$. We show that the set of such polynomials, together with the constants of $\F_{q'}$, is an $\F_{q'}$-vector space of dimension $2^{[\F_{q}:\F_{q'}]}$. Our approach not only provides the exact number of such polynomials, but also yields a construction of new examples of minimal value set polynomials for some other fixed value sets. In the latter case, we also derive a non-trivial lower bound for the number of such polynomials.

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