R-fat linearized polynomials over finite fields

Abstract

r-fat polynomials are a natural generalization of scattered polynomials. They define linear sets of the projective line PG(1,qn) of rank n with r points of weight larger than one. Using techniques on algebraic curves and function fields, we obtain numerical bounds for r and the non-existence of exceptional r-fat polynomials with r>0. We completely determine the possible values of r when considering linearized polynomials over Fq4 and we also provide one family of 1-fat polynomials in PG(1,q5). Furthermore, we investigate LP-polynomials (i.e. polynomials of type f(x)=x+δxq2s∈Fqn[x], gcd⁡(n,s)=1), determining the spectrum of values r for which such polynomials are r-fat.