Square-free values of multivariate polynomials over function fields in linear sparse sets

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Let [Formula: see text] be a square-free polynomial where [Formula: see text] is a field of [Formula: see text] elements. We view [Formula: see text] as a polynomial in the variable [Formula: see text] with coefficients in the ring [Formula: see text]. We study square-free values of [Formula: see text] in sparse subsets of [Formula: see text] which are given by a linear condition. The motivation for our study is an analogue problem of representing square-free integers by integer polynomials, where it is conjectured that setting aside some simple exceptional cases, a square-free polynomial [Formula: see text] takes infinitely many square-free values. Let [Formula: see text] be co-prime to [Formula: see text], and let [Formula: see text]. A consequence of the main result we show is that if [Formula: see text] is sufficiently large with respect to [Formula: see text] and [Formula: see text], then there exist [Formula: see text] such that [Formula: see text] is square-free. Moreover, as [Formula: see text], the last is true for almost all [Formula: see text]. The main result shows that a similar result holds also for other cases. We then generalize the results to multivariate polynomials.