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, Gurvich recently introduced a generalization mex_b 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 recursion: (Formula presented.), and conjectured that for all a, b the limits ℓ(a, b) = x _n(a, b)/n exist and are irrational algebraic numbers. In this paper we prove that showing that ℓ(a,b) = a/r-1, where r > 1 is the Perron root of the polynomial(Formula presented.) whenever a and b are coprime; furthermore, it is known that ℓ(ka, kb) = kℓ(a, b). In particular, ℓ(a, 1) = α_a = 1/2 (2 - a + √a² + 4). In 1982, Fraenkel introduced the game NIM(a) = NIM(a, 1), obtained the above recursion and solved it explicitly getting x_n = ⌊ α_a n⌊, y_n = x_n + an = ⌊ (α_a + a) n ⌊. Here we provide a polynomial time algorithm based on the Perron-Frobenius theory solving game NIM(a, b), although we have no explicit formula for its kernel.