NOTES ON GENERALISED INTEGRAL POLYNOMIAL PELL EQUATIONS

Determining the polynomials $D \in {\mathbb Z}[x]$ such that the polynomial Pell equation ${P^2-DQ^2=1}$ has nontrivial solutions $P,Q$ in ${\mathbb Q}[x]$ (and in ${\mathbb Z}[x]$ ) is an open question. In this article, we consider the generalized polynomial Pell equation $P^2-DQ^2=n$ , where $D \in {\mathbb Z}[x]$ is a monic quadratic polynomial and n is a nonzero integer. For $n=1$ , such an equation always has nontrivial solutions in ${\mathbb Q}[x]$ , but for a non-square integer n, the generalized polynomial Pell equation $P^2-DQ^2=n$ may not always have a solution in ${\mathbb Q}[x]$ . Depending on n, we determine the polynomials $D=x^2+cx+d$ , for which the equation $P^2-DQ^2=n$ has nontrivial solutions in ${\mathbb Q}[x]$ and in ${\mathbb Z}[x]$ . Taking $n=-1$ , this allows us to solve the negative polynomial Pell equation completely for any such D. An interesting feature is that there are certain polynomials D for which the generalized polynomial Pell equation has nontrivial solutions in ${\mathbb Z}[x]$ , but only finitely many, whereas the solutions in ${\mathbb Q}[x]$ are infinitely many. Finally, we determine the monic quadratic polynomials D for which the solutions of $P^2-DQ^2=n$ in ${\mathbb Z}[x]$ exhibit this finiteness phenomenon.

Solutions of polynomial Pell's equation

Polynomial Pell's equation–II

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.

Single-Axis, Spin-to-Spin Slew Maneuvers Under a Finite Jerk Constraint

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.

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.

For each positive integer $n$ it is shown how to construct a finite\ncollection of multivariable polynomials $\\{F_{i}:=F_{i}(t,X_{1},..., X_{\\lfloor\n\\frac{n+1}{2} \\rfloor})\\}$ such that each positive integer whose squareroot has\na continued fraction expansion with period $n+1$ lies in the range of exactly\none of these polynomials. Moreover, each of these polynomials satisfy a\npolynomial Pell's equation $C_{i}^{2} -F_{i}H_{i}^{2} = (-1)^{n-1}$ (where\n$C_{i}$ and $H_{i}$ are polynomials in the variables $t,X_{1},..., X_{\\lfloor\n\\frac{n+1}{2} \\rfloor}$) and the fundamental solution can be written down.\nLikewise, if all the $X_{i}$'s and $t$ are non-negative then the continued\nfraction expansion of $\\sqrt{F_{i}}$ can be written down. Furthermore, the\ncongruence class modulo 4 of $F_{i}$ depends in a simple way on the variables\n$t,X_{1},..., X_{\\lfloor \\frac{n+1}{2} \\rfloor}$ so that the fundamental unit\ncan be written down for a large class of real quadratic fields. Along the way a\ncomplete solution is given to the problem of determining for which symmetric\nstrings of positive integers $a_{1},..., a_{n}$ do there exist positive\nintegers $D$ and $a_{0}$ such that $\\sqrt{D} = [ a_{0};\\bar{a_{1}, &gt;...,\na_{n},2a_{0}}]$.\n

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.

We 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.

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.

We study the geometry of algebraic numbers in the complex plane, and their Diophantine approximation, aided by extensive computer visualization. Motivated by the resulting images, which we have called algebraic starscapes, we describe the geometry of the map from the coefficient space of polynomials to the root space, focusing on the quadratic and cubic cases. The geometry describes and explains the notable features of the illustrations, and motivates a geometric-minded recasting of fundamental results in the Diophantine approximation of the complex plane. Meanwhile, the images provide a case-study in the symbiosis of illustration and research, and an entry-point to geometry and number theory for a wider audience. In particular, the paper is written to provide an accessible introduction to the study of homogeneous geometry and Diophantine approximation. We investigate the homogeneous geometry of root and coefficient spaces under the natural action. Hyperbolic geometry and the discriminant play an important role in low degree. In particular, we rediscover the quadratic and cubic root formulas as isometries of and its unit tangent bundle, respectively. Utilizing this geometry, we determine when the map sending certain families of polynomials to their complex roots (our starscape images) are embeddings. We reconsider the fundamental questions of the Diophantine approximation of complex numbers by algebraic numbers of bounded degree, from the geometric perspective developed. In the quadratic case (approximation by quadratic irrationals), we consider approximation in terms of hyperbolic distance between roots in the complex plane and the discriminant as a measure of arithmetic height on a polynomial. In particular, we determine the supremum on the exponent k for which an algebraic target α has infinitely many approximations β whose hyperbolic distance from α does not exceed . It turns out to fall into two cases, depending on whether α lies on the image of a plane of rational slope in coefficient space (a rational geodesic). The result comes as an application of Schmidt’s subspace theorem. Our results recover the quadratic case of results of Bugeaud and Evertse, and give some geometric explanation for the dichotomy they discovered. Our statements go a little further in distinguishing approximability in terms of whether the target or approximations lie on rational geodesics. The paper comes with accompanying software, and finishes with a wide variety of open problems.

Genus one curves defined by separated variable polynomials and a 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.

Experimentation and the evaluation of algorithms have a long history in algebra. In this paper we follow a single test example over more than 250 years. In 1685, John Wallis published A treatise of algebra, both historical and practical, containing a solution of Colonel Titus’ problem that was proposed to him around 1650. The Colonel Titus problem consists of three algebraic quadratic equations in three unknowns, which Wallis transformed into the problem of finding the roots of a fourth-order (quartic) polynomial. When Joseph Raphson published his method in 1690, he demonstrated the method on 32 algebraic equations and one of the examples was this quartic equation. Edmund Halley later used the same polynomial as an example for his new methods in 1694. Although Wallis used the method of Vietè, which is a digit–by–digit method, the more efficient methods of Halley and Raphson are clearly demonstrated in the works by Raphson and Halley. For more than 250 years the quartic equation has been used as an example in a wide range of solution methods for nonlinear equations. This paper provides an overview of the Colonel Titus problem and the equation first derived by Wallis. The quartic equation has four positive roots and the equation has been found to be very useful for analyzing the number of roots and finding intervals for the individual roots, in the Cardan–Ferrari direct approach for solving quartic equations, and in Sturm’s method of determining the number of real roots of an algebraic equation. The quartic equation, together with two other algebraic equations, have likely been the first set of test examples used to compare different iteration methods of solving algebraic equations.