Polynomial Pell Equation Research Articles

Some new results on the negative polynomial Pell’s equation

We consider the negative polynomial Pell's equation $P^2(X)-D(X)Q^2(X)=-1$, where $D(X)\in \mathbb{Z}[X]$ be some fixed, monic, square-free, even degree polynomials. In this paper, we investigate the existence of polynomial solutions $P(X), Q(X)$ with integer coefficients.

Betti maps, Pell equations in polynomials and almost-Belyi maps

AbstractWe study the Betti map of a particular (but relevant) section of the family of Jacobians of hyperelliptic curves using the polynomial Pell equation$A^2-DB^2=1$, with$A,B,D\in \mathbb {C}[t]$and certain ramified covers$\mathbb {P}^1\to \mathbb {P}^1$arising from such equation and having heavy constrains on their ramification. In particular, we obtain a special case of a result of André, Corvaja and Zannier on the submersivity of the Betti map by studying the locus of the polynomialsDthat fit in a Pell equation inside the space of polynomials of fixed even degree. Moreover, Riemann existence theorem associates to the abovementioned covers certain permutation representations: We are able to characterize the representations corresponding to ‘primitive’ solutions of the Pell equation or to powers of solutions of lower degree and give a combinatorial description of these representations whenDhas degree 4. In turn, this characterization gives back some precise information about the rational values of the Betti map.

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

The polynomial Pell equation is $$\begin{aligned} P^2 - D Q^2 = 1, \end{aligned}$$ where D is a given integer polynomial and the solutions P, Q must be integer polynomials. A classical paper of Nathanson (Proc Am Math Soc 86:89–92, 1976) solved it when $$D(x) = x^2 + d$$ . We show that the Redei polynomials can be used in a very simple and direct way for providing these solutions. Moreover, this approach allows us to find all the integer polynomial solutions when $$D(x) = f^2(x) + d$$ , for any $$f \in {\mathbb {Z}}[X]$$ and $$d \in {\mathbb {Z}}$$ , generalizing the result of Nathanson. We are also able to find solutions of some generalized polynomial Pell equations introducing an extension of Redei polynomials to higher degrees.

Unlikely intersections in families of abelian varieties and the polynomial Pell equation

Let S be a smooth irreducible curve defined over a number field k and consider an abelian scheme A over S and a curve C inside A, both defined over k. In previous works, we proved that when A is a fibered product of elliptic schemes, if C is not contained in a proper subgroup scheme of A, then it contains at most finitely many points that belong to a flat subgroup scheme of codimension at least 2. In this article, we continue our investigation and settle the crucial case of powers of simple abelian schemes of relative dimension g bigger or equal than 2. This, combined with the above mentioned result and work by Habegger and Pila, gives the statement for general abelian schemes. These results have applications in the study of solvability of almost-Pell equations in polynomials.

Orthogonal-type polynomials and Pell equations

We study solutions of higher degree analogues of the polynomial Pell equation. To do so, we introduce a weight function defined on a set of line segments in the complex plane. From the moments with respect to this weight function, we construct a sequence of polynomials in a similar construction to that of orthogonal polynomials. We show that polynomials of this type appear as solutions of higher order polynomial Pell equations. We give explicit formulae and derive a set of recurrence relations for these polynomials. We also formulate a system of matrix-valued Riemann–Hilbert problems in analogy to the problem for orthogonal polynomials and show that the polynomial solutions to the Pell equations appear in solutions to such Riemann–Hilbert problems.

The polynomial Pell equation

Artūras Dubickas received his Ph.D. from the Moscow State University in 1990. He then had a postdoctoral position at the University of Witwatersrand in Johannesburg. Since 1994 he has a position at Vilnius University (Lithuania), where he is now chief research fellow and professor in the Department of Mathematics and Informatics. His main fields of research are algebraic number theory and transcendence theory.

Polynomial Pell's equation–II

The polynomial Pell's equation is X 2− DY 2=1, where D is a polynomial with integer coefficients and the solutions X, Y must be polynomials with integer coefficients. Let D= A 2+2 C be a polynomial in Z[x] , where deg C< deg A . Then for pB=pA/C∈ Z[x],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.

Multi-variable polynomial solutions to Pell’s equation and fundamental units in real quadratic fields

For each positive integer $n$ it is shown how to construct a finite collection of multivariable polynomials $\{F_{i}:=F_{i}(t,X_{1},..., X_{\lfloor \frac{n+1}{2} \rfloor})\}$ such that each positive integer whose squareroot has a continued fraction expansion with period $n+1$ lies in the range of exactly one of these polynomials. Moreover, each of these polynomials satisfy a polynomial Pell's equation $C_{i}^{2} -F_{i}H_{i}^{2} = (-1)^{n-1}$ (where $C_{i}$ and $H_{i}$ are polynomials in the variables $t,X_{1},..., X_{\lfloor \frac{n+1}{2} \rfloor}$) and the fundamental solution can be written down. Likewise, if all the $X_{i}$'s and $t$ are non-negative then the continued fraction expansion of $\sqrt{F_{i}}$ can be written down. Furthermore, the congruence class modulo 4 of $F_{i}$ depends in a simple way on the variables $t,X_{1},..., X_{\lfloor \frac{n+1}{2} \rfloor}$ so that the fundamental unit can be written down for a large class of real quadratic fields. Along the way a complete solution is given to the problem of determining for which symmetric strings of positive integers $a_{1},..., a_{n}$ do there exist positive integers $D$ and $a_{0}$ such that $\sqrt{D} = [ a_{0};\bar{a_{1}, >..., a_{n},2a_{0}}]$.

Polynomial Pell’s equation

Consider the polynomial Pell's equation X 2 - DY 2 = 1, where D = A 2 + 2C is a monic polynomial in Z[x] and deg C < degA. Then for A,C ∈ Q[x], degC < 2, and B = A/C ∈ Q[x], a necessary and sufficient condition for the polynomial Pell's equation to have a nontrivial solution in Z[x] is obtained.

Genus one curves defined by separated variable polynomials and a polynomial Pell equation

Genus one curves defined by separated variable polynomials and a polynomial Pell equation