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Valiant introduced 20 years ago an algebraic complexity theory to study the complexity of polynomial families. The basic computation model used is the arithmetic circuit, which makes these classes very easy to define and open to combinatorial techniques. In this paper we gather known results and new techniques under a unifying theme, namely the restrictions… (More)

We study the following decision problem: is the language recognized by a quantum finite automaton empty or nonempty? We prove that this problem is decidable or undecidable depending on whether recognition is defined by strict or nonstrict thresholds. This result is in contrast with the corresponding situation for probabilistic finite automata, for which it… (More)

Simon in his FOCS'94 paper was the first to show an exponential gap between classical and quantum computation. The problem he dealt with is now part of a well-studied class of problems, the hidden subgroup problems. We study Simon's problem from the point of view of quantum query complexity and give here a first nontrivial lower bound on the query… (More)

The multivariate resultant is a fundamental tool of computational algebraic geometry. It can in particular be used to decide whether a system of n homogeneous equations in n variables is satisfiable (the resultant is a polynomial in the system's coefficients which vanishes if and only if the system is satisfiable). In this paper we present several… (More)

Polynomial identity testing and arithmetic circuit lower bounds are two central questions in algebraic complexity theory. It is an intriguing fact that these questions are actually related. One of the authors of the present paper has recently proposed a " real τ-conjecture " which is inspired by this connection. The real τ-conjecture states that the number… (More)

According to the real τ-conjecture, the number of real roots of a sum of products of sparse univariate 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. In this paper, we use the Wronksian determinant to… (More)

We present an algorithm which computes the multilinear factors of bivariate lacunary polynomials. It is based on a new Gap theorem which allows to test whether <i>P(X)</i>=∑<sup><i>k</i></sup><sub><i>j</i>=1</sub> α<sub><i>j</i></sub><i>X</i><sup>α<i>j</i></sup>(1+<i>X</i>)<sup><i>βj</i></sup>is identically zero in polynomial time. The… (More)

We present general methods for proving lower bounds on the query complexity of nonadaptive quantum algorithms. Our results are based on the adversary method of Ambainis.

Simon in his FOCS'94 paper was the first to show an exponential gap between classical and quantum computation. The problem he dealt with is now part of a well-studied class of problems, the hidden subgroup problems. We stu<dy Simon's problem from the point of view of quantum query complexity and give here a first nontrivial lower bound on the query… (More)

We present a deterministic polynomial-time algorithm which computes the multilinear factors of multivariate lacunary polynomials over number fields. It is based on a new Gap theorem which allows to test whether P(X) = ∑ k j=1 a j X α j (vX + t) β j (uX + w) γ j is identically zero in polynomial time. Previous algorithms for this task were based on Gap… (More)