Event № 131
Let f be a square-free polynomial in Fq[t][x] where Fq is a field of qelements. We view f as a univariate polynomial in x with coefficientsin the ring Fq[t]. We study square-free values of f in sparse subsetsof Fq[t] which are given by a linear condition. The motivation for ourstudy is an analogue problem of representing square-free integers byinteger polynomials, where it is conjectured that setting aside somesimple exceptional cases, a square-free polynomial f in Z[x] takesinfinitely many square-free values. Let c(t) be an arbitrarypolynomial in Fq[t]. A consequence of the main result we show, is thatif q is sufficiently large with respect to the degree of c(t) and thedegrees of f in t and x, then there exist v,w in Fq such thatf(t,c(t)+vt+w) is square-free, i.e. a square-free value of f isobtained by varying the first two coefficients of c(t).