The Exact Order of Mixing in a Problem of Ward: A Seven-Term Polynomial with No Sparser Multiple

Problem A of T. Ward's list "Six problems in algebraic dynamics" concerns the polynomial f = 1 + u_1 u_2 + u_1^2 u_2 + u_1^3 u_2 + u_1^4 + u_2^2 + u_1^4 u_2^2 and the algebraic Z^2-action attached to the module Z[u_1^{±1}, u_2^{±1}]/⟨p, f⟩, where p is a prime for which f is irreducible modulo p. Its order of mixing M was known to satisfy 3 ≤ M < 7, and the problem asks for the exact value. We show that M = 6 for every prime p: the action is mixing of order 6 and not of order 7. The polynomial f is absolutely irreducible modulo every prime, so no prime has to be excluded. By results of Schmidt, Masser, and Derksen and Masser, and because the radical of the group generated by u_1, u_2 in the function field of the curve f = 0 consists only of constant multiples of monomials, M + 1 is the least number of terms of a non-zero multiple of f in F_p[u_1^{±1}, u_2^{±1}]. Our main result is that every non-zero multiple of f with coefficients in an algebraic closure of F_p has at least seven terms. The proof is elementary: it sorts a multiple by powers of u_2, reduces to two possible shapes of a six-term multiple by a Chebyshev-type recurrence, and excludes these by a local analysis at the points of the curve with u_1^4 = −1, a Frobenius splitting and an explicit computation. The prime 2 is treated separately. For p = 2 an exhaustive computer search confirms the main result for coefficients in F_2, independently of this proof, and shows that the support of f is, up to equivalence, the only non-mixing set with seven points. This is an unrefereed note.