(The published version is available online.)

Let T_n(x) denote the degree-n Chebyshev polynomial (of the first kind). This polynomial has integer coefficients, so c → T_n(c) defines a map from any field K to itself. If K is finite then there are only finitely many self-mappings of K, so there must be infinitely many pairs of distinct Chebyshev polynomials which induce the same map K→K. We show that, if K is a finite field whose cardinality q is odd, then T_n and T_m induce the same self-mapping of K if and only if n is congruent to either ±m or ±qm modulo (q²-1)/2. Further, the number of self-mappings of K induced by Chebyshev polynomials is (q+1)(q+3)/8. Finally, since T_n o T_m = T_mn, the set of self-mappings of K induced by Chebyshev polynomials is closed under composition, and hence forms a semigroup; we determine the structure of this semigroup.

These questions arose as an offshoot of the more general problem of understanding the structure (or even just the size!) of the semigroup of maps from a finite set to itself generated by a handful of maps. For instance, one can ask what is the expected size of the semigroup of maps Z/nZ→Z/nZ generated by two random maps? Preliminary calculations pointed out certain pairs of maps for which this semigroup is unusually small, and closer inspection revealed that these pairs of maps were induced by either Chebyshev polynomials or related polynomials. This motivated our detailed analysis of the semigroup of Chebyshev mappings of finite fields.

Michael Zieve:  home page   publication list