A polynomial algorithm for a two parameter extension of Wythoff NIM based on the Perron-Frobenius theory

For any positive integer parameters a and b, the second author recently introduced a generalization mexb of the standard minimum excludant mex = mex1, along with a game NIM(a, b) that extends further Fraenkel’s NIM = NIM(a, 1), which in its turn is a generalization of the classical Wythoff NIM = NIM(1, 1). It was shown that P-positions (the kernel) of NIM(a, b) are given by the following typical recursion: xn = mexb{xi, yi | 0 ≤ i < n}, yn = xn + an; n ≥ 0, and conjectured that for all a, b the limits `(a, b) = xn(a, b)/n exist and are irrational algebraic numbers. In this paper we prove this conjecture showing that `(a, b) = a r−1 , where r > 1 is the Perron root of the polynomial P (z) = z − z − 1− a−1 ∑