Indefinite quadratic polynomials of small signature
type: Item
instance of: scholarly article
title: Indefinite quadratic polynomials of small signature
zbMATH Open document ID: 0535.10022
DOI: 10.1007/BF01299144
author: N. E. Zubov
published in: Monatshefte für Mathematik
publication date: 1984-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/178172
review text: Let Q(x) be an indefinite quadratic form in n variables, then Q may be expressed as a sum of squares of real linear forms. Suppose that there are r positive signs and n-r negative signs with \(\min(r,n-r)\geq k.\) Theorem. Let \(F(x)=Q(x)+L(x)\) be a polynomial having no constant term. For any \(\eta>0\) and all large X there is an integer vector \(x\neq 0\) satisfying \(| x| \leq X\) and \(| F(x)|<<X^{f(n,k)+\eta}\) where \[ f(n,k)=-1/2+3/(4k+2)+O_ k(1/n)\quad \to \quad -1/2\quad as\quad n\to \infty,\quad k\to \infty. \] The analogous results for Hermitian forms and polynomials in the order \(x+y\sqrt{d}\), \(d<0\) square- free, are also obtained.
Mathematics Subject Classification ID: 11E16
Mathematics Subject Classification ID: 11D75
zbMATH DE Number: 3849325
zbMATH Keywords: indefinite quadratic polynomials
zbMATH Keywords: small signature
zbMATH Keywords: indefinite quadratic form
zbMATH Keywords: sum of squares of real linear forms
zbMATH Keywords: Hermitian forms
MaRDI profile type: Publication
cites work: On a theorem of Davenport and Heilbronn
cites work: Indefinite quadratic forms in many variables
cites work: Indefinite Hermitian Forms
cites work: Indefinite quadratic polynomials
cites work: Q4723815
OpenAlex ID: W2029903851
description: scientific article; zbMATH DE number 3849325