Weyl sums in Fq[x] with digital restrictions

AbstractLet Fq be a finite field and consider the polynomial ring Fq[X]. Let Q∈Fq[X]. A function f:Fq[X]→G, where G is a group, is called strongly Q-additive, if f(AQ+B)=f(A)+f(B) holds for all polynomials A,B∈Fq[X] with degB<degQ. We estimate Weyl sums in Fq[X] restricted by Q-additive functions. In particular, for a certain character E we study sums of the form∑PE(h(P)), where h∈Fq((X−1))[Y] is a polynomial with coefficients contained in the field of formal Laurent series over Fq and the range of P is restricted by conditions on fi(P), where fi (1⩽i⩽r) are Qi-additive functions. Adopting an idea of Gel'fond such sums can be rewritten as sums of the form∑degP