The Smallest 2-Pisot Numbers in Fq((X-1)) Where q is Different from the Power of 2

The aim of this paper is to characterize the smallest 2-Pisot number of degree n > 2 in the case of Laurent power series Fq((X-1)) with q distinct from the power of 2 over a finite field Fq by giving explicitly its minimal polynomial. Indeed, we prove that its minimal polynomial is given by ?n(Y ) = Yn - & alpha;X(X + 1)Y n-1 - & alpha;2X3Y n-2 +& alpha;n. We show in particular that the sequence of smallest 2-pisot numbers of degree n is decreasing and converges to (& alpha;X2, & alpha;X) where we suppose that & alpha; is the least element of Fq \ {0}.

Automated summary

The aim of this paper is to characterize the smallest 2-Pisot number of degree n > 2 in the case of Laurent power series Fq((X-1)) with q distinct from the power of 2 over a finite field Fq by giving explicitly its minimal polynomial. Indeed, we prove that its minimal polynomial is given by ?n(Y ) = Yn - αX(X + 1)Yn-1 - α2X3Yn-2 + αn. We show in particular that the sequence of smallest 2-pisot numbers of degree n is decreasing and converges to (αX2, αX) where we suppose that α is the least element of Fq \ {0}.