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# Small Zeros of Quadratic Forms Avoiding a Finite Number of Prescribed Hyperplanes

Published:2009-03-01
Printed: Mar 2009
• Rainer Dietmann
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## Abstract

We prove a new upper bound for the smallest zero $\mathbf{x}$ of a quadratic form over a number field with the additional restriction that $\mathbf{x}$ does not lie in a finite number of $m$ prescribed hyperplanes. Our bound is polynomial in the height of the quadratic form, with an exponent depending only on the number of variables but not on $m$.
 MSC Classifications: 11D09 - Quadratic and bilinear equations 11E12 - Quadratic forms over global rings and fields 11H46 - Products of linear forms 11H55 - Quadratic forms (reduction theory, extreme forms, etc.)