Polynomial Pell's equation–II

Abstract

The polynomial Pell's equation is X 2 À DY 2 ¼ 1; where D is a polynomial with integer coefficients and the solutions X ; Y must be polynomials with integer coefficients. Let D ¼ A 2 þ 2C be a polynomial in Z½x; where deg Codeg A: Then for pB ¼ pA=CAZ½x; p a prime, a necessary and sufficient condition for which the polynomial Pell's equation has a nontrivial solution is obtained. Furthermore, all solutions to the polynomial Pell's equation satisfying the above condition are determined. r 2004 Elsevier Inc. All rights reserved.

The Solvability of Polynomial Pell’s Equation

“…Such result is generalized when in Webb and Yokota (2004), where is prime without any condition of . In this case, the authors also determined the solutions.…”

“…Then there exist infinitely many pairs 2. Thus, to determine all rational solutions of equation 2, it suffices to all solutions in in Webb and Yokota (2004). Among all solutions in T, say is a fundamental solution if and only if its non-Archimedean absolute value satisfies the condition , for all…”

ON THE EQUATION f(g(x))=f(x)hm(x) FOR COMPOSITE POLYNOMIALS

“…Then, using the method outlined in Section 1, one can easily prove the following analogue of Theorem 2 in [2] for the sign changes of λ function at rational points: either λ( f (r)) is constant for all rational numbers r greater than the largest real root of g(x) − x or it changes sign infinitely many often. The question of finding all solutions of the composition equation in integer polynomials f (x), g(x), and h(x) is closely related to the solution of the polynomial Pell equations in Z[x]; see [9,10,14]. This does not seem to be easy.…”

A note on the use of Rédei polynomials for solving the polynomial Pell equation and its generalization to higher degrees

“…Webb and Yokota [17] found a necessary and sufficient condition for which the polynomial Pell equation has non-trivial solutions when D = A 2 +2C monic polynomial, A/C ∈ Z[X ] and deg C < 2. In [18], such a result is generalized when p A/C ∈ Z[X ], for some prime p, without any condition on the degree of C. In this case, the authors also determined the solutions. Then, Yokota [19] found a necessary and sufficient condition for the solution of the polynomial Pell equation when A/C ∈ Q[X ].…”