Determinantal complexity
نویسنده
چکیده
منابع مشابه
A lower bound for the determinantal complexity of a hypersurface
We prove that the determinantal complexity of a hypersurface of degree d > 2 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 3×3 permanent is 7. We also prove that for n > 3, there is no nonsingular hypersurface in Pn of degree d that has an expression as a dete...
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Tavenas has recently proved that any nO(1)-variate and degree n polynomial in VP can be computed by a depth-4 ΣΠ[O( p n)]ΣΠ[ p n] circuit of size 2O( p n log n) [Tav13]. So to prove VP 6= VNP, it is sufficient to show that an explicit polynomial ∈ VNP of degree n requires 2ω( p n log n) size depth-4 circuits. Soon after Tavenas’s result, for two different explicit polynomials, depth-4 circuit s...
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We answer a question in [10], showing the regular determinantal complexity of the determinant detm is O(m ). We answer questions in, and generalize results of [2], showing there is no rank one determinantal expression for perm m or detm when m ≥ 3. Finally we state and prove several “folklore” results relating different models of computation.
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The permanent vs. determinant problem is one of the most important problems in theoretical computer science, and is the main target of geometric complexity theory proposed by Mulmuley and Sohoni. The current best lower bound for the determinantal complexity of the d by d permanent polynomial is d/2, due to Mignon and Ressayre in 2004. Inspired by their proof method, we introduce a natural rank ...
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We study the family of ideals defined by mixed size minors of two-sided ladders of indeterminates. We compute their Gröbner bases with respect to a skewdiagonal monomial order, then we use them to compute the height of the ideals. We show that these ideals correspond to a family of irreducible projective varieties, that we call mixed ladder determinantal varieties. We show that these varieties ...
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