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

Abstract

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:

$$x_n = {\rm mex}_b(\{x_i, y_i \;|\; 0 \leq i < n\}), \;\; y_n = x_n + an; \;\; n \geq 0,$$

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 ${\ell(a,b) = \frac{a}{r-1}}$, where r > 1 is the Perron root of the polynomial

$$P(z) = z^{b+1} - z - 1 - \sum_{i=1}^{a-1} z^{\lceil ib/a \rceil},$$

whenever a and b are coprime; furthermore, it is known that ℓ(ka, kb) = kℓ(a, b). In particular, ${\ell(a, 1) = \alpha_a = \frac{1}{2} (2 - a + \sqrt{a^2 + 4})}$. In 1982, Fraenkel introduced the game NIM(a) = NIM(a, 1), obtained the above recursion and solved it explicitly getting ${x_n = \lfloor \alpha_a n \rfloor, \; y_n = x_n + an = \lfloor (\alpha_a + a) n \rfloor}$. 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.