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Counting irreducible components of complex algebraic varieties. Accepted for Computational Complexity
, 2008
"... Abstract. We present an algorithm for counting the irreducible components of a complex algebraic variety defined by a fixed number of polynomials encoded as straightline programs (slps). It runs in polynomial time in the BlumShubSmale (BSS) model and in randomized parallel polylogarithmic time in ..."
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Abstract. We present an algorithm for counting the irreducible components of a complex algebraic variety defined by a fixed number of polynomials encoded as straightline programs (slps). It runs in polynomial time in the BlumShubSmale (BSS) model and in randomized parallel polylogarithmic time in the Turing model, both measured in the lengths and degrees of the slps. Our algorithm is obtained from an explicit version of Bertini’s theorem. For its analysis we further develop a general complexity theoretic framework appropriate for algorithms in algebraic geometry.
CastelnuovoMumford Regularity and Computing the de Rham Cohomology of Smooth Projective Varieties
, 905
"... We prove that the CastelnuovoMumford regularity of the sheaf of differential pforms on a smooth complex projective variety X is bounded by p(e + 1)D dim X, where e and D are the maximal codimension resp. degree of all irreducible components of X. This result allows us to give a parallel polynomial ..."
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We prove that the CastelnuovoMumford regularity of the sheaf of differential pforms on a smooth complex projective variety X is bounded by p(e + 1)D dim X, where e and D are the maximal codimension resp. degree of all irreducible components of X. This result allows us to give a parallel polynomial time algorithm for computing the algebraic de Rham cohomology of X. We also include a test whether a projective variety is smooth running in parallel polynomial time. 1
Randomization, Relaxation, and Complexity
, 2010
"... Systems of polynomial equations arise naturally in applications ranging from the study of chemical reactions to coding theory to geometry and number theory. Furthermore, the complexity of the equations we wish to solve continues to rise: while engineers in ancient Egypt needed to solve quadratic equ ..."
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Systems of polynomial equations arise naturally in applications ranging from the study of chemical reactions to coding theory to geometry and number theory. Furthermore, the complexity of the equations we wish to solve continues to rise: while engineers in ancient Egypt needed to solve quadratic equations in one variable, today we have applications in satellite orbit design and combustive fluid flow hinging on the solution of systems of polynomial equations involving dozens or even thousands of variables. For example, the lefthand illustration above shows an instance of the orbit transfer problem, while the righthand illustration above shows a level set for a reactive fluid flow. More precisely, in the first problem, one wants to use N blasts of a rocket to transfer a satellite from an initial orbit to a desired final orbit, using as little fuel as possible. The optimal rocket timings and directions can then be reformulated as the real solutions of a system of 45N sparse polynomial equations in 45N variables, thanks to recent work of Avendano and Mortari [1]. For reactive fluid flow, a standard technique is to decimate the space into small cubes and obtain an approximation to some parameter function (such as vorticity or temperature) via an expansion into polynomial basis functions. Asking for regions where a certain parameter lies in a certain interval then reduces to solving millions of polynomial systems — the precise number depending on the region and size of the cubes. Far from laying the subject to rest, modern hardware and software has led us to even deeper unsolved problems concerning the hardness of solving. These questions traverse not only algebraic geometry but also number theory, algorithmic complexity, numerical analysis, and probability theory. The need to look beyond computational algebra for new algorithms is thus one of the main motivations behind this workshop.