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

[摘要] We prove a new upper bound for the smallest zero $mathbf{x}$of a quadratic form over a number field with the additionalrestriction that $mathbf{x}$ does not lie in a finite number of $m$ prescribedhyperplanes. Our bound is polynomial in the height of the quadraticform, with an exponent depending only on the number of variables butnot on $m$.

[学科分类] 数学（综合）

[关键词] Ordered rings with involution;$C^ast$-algebras and their representations;noncommutative convexity theory;real algebraic geometry