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Quadratic and Higher Degree Forms

Part of the Developments in Mathematics Book Series (volume 31)

Editors: Alladi, Krishnaswami; Bhargava, Manjul; Savitt, David; Tiep, Pham Huu

DOI: 10.1007/978-1-4614-7488-3

ISBN: 978-1-4614-7487-6

Publisher: Springer New York - Springer Journals

Jul 1, 2013

1 - 10 of 10 Chapters

LitStream

Chapter 1Toy Models for D. H. Lehmer’s Conjecture II

Bannai, Eiichi; Miezaki, Tsuyoshi

Pages: 1 –27

[In the previous paper under the same title, we showed that the m-th Fourier coefficient of the weighted theta series of the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy}...
LitStream

Chapter 2On Representation of an Integer by X2 + Y2 + Z2 and the Modular Equations of Degree 3 and 5

Berkovich, Alexander

Pages: 29 –49

[I discuss a variety of results involving s(n), the number of representations of n as a sum of three squares. One of my objectives is to reveal numerous interesting connections between the properties of this function and certain modular equations of degree 3 and 5. In particular, I show that...
LitStream

Chapter 3Almost Universal Ternary Sums of Squares and Triangular Numbers

Chan, Wai Kiu; Haensch, Anna

Pages: 51 –62

[For any integer x, let Tx denote the triangular number \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek}...
LitStream

Chapter 4Weighted Generating Functions for Type II Lattices and Codes

Elkies, Noam D.; Kominers, Scott Duke

Pages: 63 –108

[We give a new structural development of harmonic polynomials on Hamming space, and harmonic weight enumerators of binary linear codes, that parallels one approach to harmonic polynomials on Euclidean space and weighted theta functions of Euclidean lattices. Namely, we use the finite-dimensional...
LitStream

Chapter 5Quadratic Forms and Automorphic Forms

Hanke, Jonathan

Pages: 109 –168

[These notes give a friendly four-part introduction to various aspects of the arithmetic and analytic theories of quadratic forms, aimed at a graduate-level audience. The main themes discussed are: geometry and local-global methods, theta functions and Siegel’s theorem, Clifford algebras and spin...
LitStream

Chapter 6Integral Positive Ternary Quadratic Forms

Jagy, William C.

Pages: 169 –179

[We discuss some families of integral positive ternary quadratic forms. Our main example is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs}...
LitStream

Chapter 7Some Aspects of the Algebraic Theory of Quadratic Forms

Parimala, R.

Pages: 181 –205

[This article, based on the lectures at the Arizona Winter School on “Quadratic forms”, gives a quick introduction to the algebraic theory of quadratic forms. It discusses some invariants associated to quadratic forms like the Pythagoras number and the u-invariant and touches on some recent...
LitStream

Chapter 8On the Length of Binary Forms

Reznick, Bruce

Pages: 207 –232

[The K-length of a form f in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}...
LitStream

Chapter 9Representation of Quadratic Forms by Integral Quadratic Forms

Schulze-Pillot, Rainer

Pages: 233 –253

[The number of representations of a positive definite integral quadratic form of rank n by another positive definite integral quadratic form of rank m ≥ n has been studied by arithmetic, analytic, and ergodic methods. We survey and compare in this article the results obtained by these methods.]
LitStream

Chapter 10Identifying the Matrix Ring: Algorithms for Quaternion Algebras and Quadratic Forms

Voight, John

Pages: 255 –298

[We discuss the relationship between quaternion algebras and quadratic forms with a focus on computational aspects. Our basic motivating problem is to determine if a given algebra of rank 4 over a commutative ring R embeds in the 2 ×2-matrix ring M2(R) and, if so, to compute such an embedding. We...

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