POLYNOMIAL VALUES AND GENERATORS WITH MISSING DIGITS IN FINITE FIELDS

We consider the linear vector space formed by the elements of the finite field F-q with q = p(r) over F-p. Then the elements x of F-q have a unique representation in the form x = Sigma(r)(j=1) c(j)a(j) with c(j) is an element of F-p; the coefficients c(j) will be called digits. Let D be a subset of F-p with 2 <= vertical bar D vertical bar < p. We consider elements x of F-q such that for their every digit c(j) we have c(j) is an element of D; then we say that the elements of F-p \ D are "missing digits". We will show that if D is a large enough subset of F-p, then there are squares with missing digits in F-q; if the degree of the polynomial f(x) is an element of F-q [X] is at least 2 then it assumes values with missing digits; there are generators g in F-q such that f (g) is of missing digits.