We extend the transfer theorem of [13] to the complex field. That is, we investigate the links be... more We extend the transfer theorem of [13] to the complex field. That is, we investigate the links between the class VPSPACE of families of polynomials and the Blum-Shub-Smale model of computation over C. Roughly speaking, a family of polynomials is in VPSPACE if its coefficients can be computed in polynomial space. Our main result is that if (uniform, constant-free) VPSPACE families can be evaluated efficiently then the class PAR C of decision problems that can be solved in parallel polynomial time over the complex field collapses to P C. As a result, one must first be able to show that there are VPSPACE families which are hard to evaluate in order to separate P C from NP C , or even from PAR C .
According to the real τ -conjecture, the number of real roots of a sum of products of sparse poly... more According to the real τ -conjecture, the number of real roots of a sum of products of sparse polynomials should be polynomially bounded in the size of such an expression. It is known that this conjecture implies a superpolynomial lower bound on the arithmetic circuit complexity of the permanent.
We deploy algebraic complexity theoretic techniques for constructing symmetric determinantal repr... more We deploy algebraic complexity theoretic techniques for constructing symmetric determinantal representations of formulas and weakly skew circuits. Our representations produce matrices of much smaller dimensions than those given in the convex geometry literature when applied to polynomials having a concise representation (as a sum of monomials, or more generally as an arithmetic formula or a weakly skew circuit). These representations are valid in any field of characteristic different from 2. In characteristic 2 we are led to an almost complete solution to a question of Bürgisser on the VNP-completeness of the partial permanent. In particular, we show that the partial permanent cannot be VNP-complete in a finite field of characteristic 2 unless the polynomial hierarchy collapses.
Mahaney's Theorem states that there are no sparse NP-complete problems if P ≠ NP.... more Mahaney's Theorem states that there are no sparse NP-complete problems if P ≠ NP. We propose an adaptation of that theorem to the Blum-Shub-Smale model of computation of the reals with addition and equality. In that setting, sparseness is defined in terms of topological dimension instead of cardinality.
We extend the transfer theorem of [13] to the complex field. That is, we investigate the links be... more We extend the transfer theorem of [13] to the complex field. That is, we investigate the links between the class VPSPACE of families of polynomials and the Blum-Shub-Smale model of computation over C. Roughly speaking, a family of polynomials is in VPSPACE if its coefficients can be computed in polynomial space. Our main result is that if (uniform, constant-free) VPSPACE families can be evaluated efficiently then the class PAR C of decision problems that can be solved in parallel polynomial time over the complex field collapses to P C. As a result, one must first be able to show that there are VPSPACE families which are hard to evaluate in order to separate P C from NP C , or even from PAR C .
According to the real τ -conjecture, the number of real roots of a sum of products of sparse poly... more According to the real τ -conjecture, the number of real roots of a sum of products of sparse polynomials should be polynomially bounded in the size of such an expression. It is known that this conjecture implies a superpolynomial lower bound on the arithmetic circuit complexity of the permanent.
We deploy algebraic complexity theoretic techniques for constructing symmetric determinantal repr... more We deploy algebraic complexity theoretic techniques for constructing symmetric determinantal representations of formulas and weakly skew circuits. Our representations produce matrices of much smaller dimensions than those given in the convex geometry literature when applied to polynomials having a concise representation (as a sum of monomials, or more generally as an arithmetic formula or a weakly skew circuit). These representations are valid in any field of characteristic different from 2. In characteristic 2 we are led to an almost complete solution to a question of Bürgisser on the VNP-completeness of the partial permanent. In particular, we show that the partial permanent cannot be VNP-complete in a finite field of characteristic 2 unless the polynomial hierarchy collapses.
Mahaney's Theorem states that there are no sparse NP-complete problems if P ≠ NP.... more Mahaney's Theorem states that there are no sparse NP-complete problems if P ≠ NP. We propose an adaptation of that theorem to the Blum-Shub-Smale model of computation of the reals with addition and equality. In that setting, sparseness is defined in terms of topological dimension instead of cardinality.
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Papers by Pascal Koiran