The polynomial Pell equation

Contributors

In this article we develop an important tool in transcendental number theory. More precisely, we study multiplicity estimates (or multiplicity lemmas) for analytic functions. Our main theorem reduces multiplicity estimates at zero to the study of ideals in polynomial ring stable under an appropriate map. In particular, in the case of algebraic morphisms this result gives a new link between the theory of polarized algebraic dynamical systems and transcendental number theory. Specialized to the case of differential operators this theorem improves Nesterenko's conditional result on solutions of systems of differential equations. We also deduce an analog of Nesterenko's theorem for Mahler's functions and for solutions of q-difference equations. Further, analyzing stable ideals we prove the unconditional optimal result in the case of linear functional systems of generalized Mahler's type. The latter result generalizes a famous theorem of Nishioka (1986) previously conjectured by Mahler (1969). This new multiplicity estimate allows to prove new results on algebraic independence and on measures of algebraic independence, as done in Zorin (2010 and 2011).

In the behavioral and social sciences, a typical source of ordered categorical data is the response of a group of people to some form of rating scale. Two frequently used rating scales are the Liken-type scale (Edwards, 1957; Likert, 1932) and the semantic differential (Osgood, Suci, & Tannenbaum, 1957). In analyzing such scales, the mean and standard deviation of the responses to each item are commonly obtained and an attempt is made to determine whether the sample respondents are or are not generally in agreement. An exact probability test for this purpose has been proposed by Cooper (1976), with important modifications suggested by Whitney (1978). The program described here calculates the Whitney t statistic for each item, together with an estimate of the probability of its occurrence (Jaspen, 1965), and it provides a nonsubjective estimate of whether or not a particular item pole is characteristic of the sample population, It may thus form a useful adjunct to existing programs for the analysis of Likert-type (Antonak, 1980) and semantic differential (Lawson & Metivier, 1980) rating scales. Input. The program reads data in the form of responses by item for each respondent. A problem parameter card is used to control for the number of item response alternatives, the number of respondents, and the number of questions. Output. The printed output provides (1) a respondent by item response matrix, (2) tables showing the number of responses for each item response alternative, together with any invalid responses, for (a) the total number in the sample, as well as for (b) the male and (c) the female subgroups in the sample, (3) the Whitney t statistic for each item, and (4) the associated probability level for each t statistic. Computer and language. The program is written in standard FORTRAN IV for use on an IBM 370/158 system. Adjustments for use with another computer will depend on the compiler used. Restrictions. The program has the following restrictions: The maximum number of item response alternatives is seven, and the maximum number of items is 150. Availability. Copies of this paper and a source listing of the program, with sample data analysis, may be obtained without charge from Michael J. Glencross, Department of Mathematics, University of the Witwatersrand, 1 Jan Smuts Avenue, Johannesburg 2001, South Africa.

The first graduation of the mathematics faculty at Gorky University happened in 1957 thanks to the support of Prof. A.A. Lyapunov. He was the head of the programming section of the Department of Mathematics at the V.A. Steklov Institute of Mathematics, USSR. He also was a professor of the mathematics and mechanics faculty of Moscow State University (MSU). The whole academic year 1956-1957 six students of the physics and mathematics faculty of GSU spent in Moscow because there were no computers in the Gorky region, including the almighty VNIIEF (Sarov).

Heigh Ho! Says Tall-y A frog he would a-wooing go Hoorah! For David O. Tall. David Tall died on 15th July, 2024 at the age of 83. He was one of the most influential members of the mathematics education community as it was establishing itself and growing through the 1980s and 90s. In particular, he was critical to establishing research in undergraduate mathematics education as a distinct subfield within a discipline previously focused on schools. Despite extensive international travel for academic work, David was a man of the Midlands in the UK, having been born and schooled in Northamptonshire, studying at Oxford and, after a short period as a lecturer at the University of Sussex, moving to Kenilworth in 1969 for a post at the then new Warwick University where he spent the rest of his long and distinguished academic career. In his early years at Warwick's Mathematics Institute, he formed a writing partnership with Ian Stewart, producing highly influential textbooks such as The Foundations of Mathematics (1977), Algebraic Number Theory (1979) (later editions incorporating new material to become Algebraic Number Theory and Fermat's Last Theorem) (2001) and Complex Analysis (1983). These are still set texts in many UK universities many decades later. David had completed a PhD on the representation of groups at Oxford, supervised by Michael Atiyah (later a Fields and Abel prize winner), but David already had a deep interest in how people think about mathematics and learn it. After moving from the Warwick Mathematics Institute to the Mathematics Education Research Centre at Warwick, he took the decision to undertake a second PhD with Richard Skemp as

In his biography of Emil Artin, Richard Brauer describes the years from 1931–1941 as a time when “Artin spoke through his students and through the members of his mathematical circle” rather than through written publications. This paper explores these seemingly quiet years when Artin immigrated to America and disseminated ideas about algebraic number theory during this time in his collaboration with George Whaples, a young American mathematician who had just completed his Ph.D. at the University of Wisconsin. The main result of their work is the use of the product formula for valuations to come up with an axiomatic characterization of both algebraic number fields and algebraic function fields with a finite field of constants. These two families of fields are exactly the fields for which class field theory is known to hold. We situate their mathematical work in the broader context of algebraic number theory and their lives within the broader historical context.

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.

n/a

Dear Reader: The Moscow Journal of Combinatorics and Number Theory was founded in 2010 by the Moscow Institute of Physics and Technology, and in 2018 it started being published by MSP (Mathematical Sciences Publishers), a nonprofit scientific publisher based in Berkeley, California. Our journal publishes original, high-quality research articles from a broad range of interests within combinatorics, number theory and allied areas. Since 2011 we have published over 100 papers. Among our authors are such mathematicians as Noga Alon, Antal Balog, Jean-Pierre Demailly, Dominic Foata, Peter Frankl, Aleksandar Ivic, Sergei Konyagin, Yuri Nesterenko, Janos Pach, Yakov Sinai, Andrzej Schinzel, Wolfgang Schmidt, Carlo Viola, Michel Waldschmidt, and many others. This issue 1 of volume 8 is the first issue to appear under MSP's aegis. It contains selected papers presented by the participants of the Vilnius Conference in Combinatorics and Number Theory, which was organized with support from our journal and took place at the Department of Mathematics and Informatics of the University of Vilnius, Lithuania, 16–22 July 2017. Previous conferences connected to our journal were held in Russia (Diophantine analysis, Astrakhan, 30 July to 3 August 2012), Lithuania (Palanga Conference in Combinatorics and Number Theory, 1–7 September 2013), again Russia (Moscow Workshop in Combinatorics and Number Theory, January 27 to 2 February 2014), and Denmark (Diophantine Approximation and Related Topics, Aarhus, 13–17 July 2015). A collection of papers from the Astrakhan conference appeared in issue 3–4 of volume 3 (2013), and papers related to the Aarhus conference in issue 2–3 of volume 6 (2016). We hope that you will enjoy this issue and support the journal both with your submissions and by recommending a subscription to your institutional library!

Contributors

This book is a translation into English of Hilbert's Theorie der algebraischen Zahlkorper best known as the Zahlbericht, first published in 1897, in which he provided an elegantly integrated overview of the development of algebraic number theory up to the end of the nineteenth century. The Zahlbericht provided also a firm foundation for further research in the subject. It is based on the work of the great number theorists of the nineteenth century. The Zahlbericht can be seen as the starting point of all twentieth century investigations in algebraic number theory, reciprocity laws and class field theory. For this English edition an Introduction has been added by F. Lemmermeyer and N. Schappacher.

PurposeThe authors have determined whether the points fixed by all the full and the partial Atkin–Lehner involutions WQ on X0(N) for N ≤ 50 are Weierstrass points or not.Design/methodology/approachThe design is by using Lawittes's and Schoeneberg's theorems.FindingsFinding all Weierstrass points on X0(N) fixed by some Atkin–Lehner involutions. Besides, the authors have listed them in a table.Originality/valueThe Weierstrass points have played an important role in algebra. For example, in algebraic number theory, they have been used by Schwartz and Hurwitz to determine the group structure of the automorphism groups of compact Riemann surfaces of genus g ≥ 2. Whereas in algebraic geometric coding theory, if one knows a Weierstrass nongap sequence of a Weierstrass point, then one is able to estimate parameters of codes in a concrete way. Finally, the set of Weierstrass points is useful in studying arithmetic and geometric properties of X0(N).

A global approach to phase calibration is presented. The corresponding theoretical framework calls on elementary concepts of algebraic graph theory (spanning tree of maximal weight, cycles) and algebraic number theory (lattice, nearest lattice point). The traditional approach can thereby be better understood. In radio imaging and in optical interferometry, the self-calibration procedures must often be conducted with much care. The analysis presented should then help in finding a better compromise between the coverage of the calibration graph (which must be as complete as possible) and the quality of the solution (which must of course be reliable).

To put the issues in a broader context, these three number-theoretic problems were instrumental in the emergence of algebraic number theory-one of the two main sources of the modern discipline of commutative algebra.' The other source was algebraic geometry. It was in the setting of these two subjects that many of the main concepts and results of commutative algebra evolved in the 19th century. Thus commutative algebra can be said to have been well developed before it was created. To set the scene for our story, a word about mathematics, and especially algebra, in the 19th century. The period witnessed! fundamental transformations in mathematics -in its concepts, its methods, and in mathematicians' attitude toward their subject. These included a growing insistence on rigor and abstraction; a predisposition for founding general theories rather than focusing on specific problems; an acceptance of nonconstructive existence proofs; the emergence of mathematical specialties and specialists; the rise of the view of mathematics as a free human activity, neither deriving from, nor dependent on, nor necessarily applicable to concrete setting; the rise of set-theoretic thinking; and the reemergence of (a new variant of) the axiomatic method. As for algebra, its subject-matter and methods changed beyond recognition. Algebra as the study of solvability of polynomial equations gave way to algebra as the study of abstract structures defined axiomatically. While in the past algebra was on the periphery of mathematics, it now (the 19th and early 20th centuries) became one of its central concerns. Moreover, algebra began to penetrate other mathematical fields (e.g. geometry, analysis, logic, topology, number theory) to such an extent that in the early decades of the 20th century one began to speak of the algebraization of mathematics ([3, p. 135] and [22]).

Recently, several intriguing conjectures have been proposed connecting symmetric informationally complete quantum measurements (SIC POVMs, or SICs) and algebraic number theory. These conjectures relate the SICs to their minimal defining algebraic number field. Testing or sharpening these conjectures requires that the SICs are expressed exactly, rather than as numerical approximations. While many exact solutions of SICs have been constructed previously using Gröbner bases, this method has probably been taken as far as is possible with current computer technology (except in special cases where there are additional symmetries). Here, we describe a method for converting high-precision numerical solutions into exact ones using an integer relation algorithm in conjunction with the Galois symmetries of an SIC. Using this method, we have calculated 69 new exact solutions, including nine new dimensions, where previously only numerical solutions were known—which more than triples the number of known exact solutions. In some cases, the solutions require number fields with degrees as high as 12 288. We use these solutions to confirm that they obey the number-theoretic conjectures, and address two questions suggested by the previous work.