科研之友
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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.
- 出版日期2017-6
