Prime power polynomial maps over finite fields

Details Prime power polynomial maps over finite fields Prime power polynomial maps over finite fields

2012-01-05

arxiv.org/abs/1201.1137v1

Mathematics

Commutative Algebra

18 pages

Scientific paper

We consider polynomial maps described by so-called prime power polynomials. These polynomials are defined using a fixed power of a prime number, say q. Considering invertible polynomial maps of this type over a characteristic zero field, we will only obtain (up to permutation of the variables) triangular maps, which are the most basic examples of polynomial automorphisms. However, over the finite field F_q automorphisms of this type have (in general) an entirely different structure. Namely, we will show that the prime power polynomial maps over F_q are in one-to-one correspondence with matrices having coefficients in a univariate polynomial ring over F_q. Furthermore, composition of polynomial maps translates to matrix multiplication, implying that invertible prime power polynomial maps correspond to invertible matrices. This alternate description of the prime power polynomial automorphism subgroup leads to the solution of many famous conjectures for this kind of polynomials and polynomial maps.

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