For a partial function $f: X \to F_q$, where $X \subseteq F_q$ be a subset of a finite field $F_q$, is there any criterions to judge whether $f$ is injective?
For $X=F_q$, since any function from $F_q$ to $F_q$ is a polynomial function, such topic has being discussed using the theory of permutation polynomials; Moreover, criterions only test on the cardinality of $deg f$ and $f(F_q)$ was given by Williams, Daqing Wan and Turnwald
For $X\subsetneq F_q$ case, one can still discuss the problem by considering the polynomials given by Newton Interpolation Formula, is there any similar way to test the injectivity of $f$ only by testing the cardinality of $X$ and $f(X)$?
References:
Williams, K. S., On exceptional polynomials, Can. Math. Bull. 11, 279-282 (1968). ZBL0159.05304.
Wan, Daqing, A $p$-adic lifting lemma and its applications to permutation polynomials, Mullen, Gary L. (ed.) et al., Finite fields, coding theory, and advances in communications and computing. Proceedings of the international conference on finite fields, coding theory, and advances in communications and computing, held at the University of Nevada, Las Vegas, USA, August 7-10, 1991. New York: Marcel Dekker, Inc. Lect. Notes Pure Appl. Math. 141, 209-216 (1993). ZBL0792.11049.
Turnwald, Gerhard, A new criterion for permutation polynomials, Finite Fields Appl. 1, No. 1, 64-82 (1995). ZBL0817.11055.