A Lower Bound for the Determinantal Complexity of a Hypersurface

被引：4

作者：

Alper, Jarod ^{[1
]}

Bogart, Tristram ^{[2
]}

Velasco, Mauricio ^{[2
]}

机构：

[1] Australian Natl Univ, Inst Math Sci, Canberra, ACT 0200, Australia

[2] Univ Los Andes, Dept Matemat, Cra 1 18A-10, Bogota 111711, Colombia

来源：

FOUNDATIONS OF COMPUTATIONAL MATHEMATICS | 2017年 / 17卷 / 03期

基金：

澳大利亚研究理事会;

关键词：

Determinantal complexity; Affine linear projections; Permanents; Cubic surfaces; PERMANENT;

D O I：

10.1007/s10208-015-9300-x

中图分类号：

TP301 [理论、方法];

学科分类号：

081202 ;

摘要：

We prove that the determinantal complexity of a hypersurface of degree is bounded below by one more than the codimension of the singular locus, provided that this codimension is at least 5. As a result, we obtain that the determinantal complexity of the 3 x 3 permanent is 7. We also prove that for n > 3 there is no nonsingular hypersurface in p(n) of degree d x d that has an expression as a determinant of a matrix of linear forms, while on the other hand for n <= 3, a general determinantal expression is nonsingular. Finally, we answer a question of Ressayre by showing that the determinantal complexity of the unique (singular) cubic surface containing a single line is 5.

引用

页码：829 / 836

页数：8

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