Papers by Peter Buergisser
We determine the expected curvature polynomial of random real projective varieties given as the z... more We determine the expected curvature polynomial of random real projective varieties given as the zero set of independent random polynomials with Gaussian distribution, whose distribution is invariant under the action of the orthogonal group. In particular, the expected Euler characteristic of such random real projective varieties is found. This considerably extends previously known results on the number of roots, the
Page 1. Reviewers Dimitris Achlioptas Micah Adler Pankaj Agarwal Dorit Aharonov Noga Alon Sriniva... more Page 1. Reviewers Dimitris Achlioptas Micah Adler Pankaj Agarwal Dorit Aharonov Noga Alon Srinivas Aluru Andris Ambainis Matthew Andrews Aaron Archer V. Arvind Hagit Attiya Yonatan Aumann Baruch Awerbuch Yossi Azar Gil Barequet Amotz Bar-Noy Sasha Barvinok Reuven Bar-Yehuda Amos Beimel Michael Bender Eli Ben-Sasson Marshall Bern Nayantara Bhatnagar Somenath Biswas Avrim Blum Nader Bshouty Peter Buergisser Harry Buhrman Lynn Burroughs Sam Buss Matt Cary Nicolo Cesa-Bianchi Amit Chakrabarti Moses Charikar ...
Corr, 2009
We discuss the geometry of orbit closures and the asymptotic behavior of Kronecker coefficients i... more We discuss the geometry of orbit closures and the asymptotic behavior of Kronecker coefficients in the context of the Geometric Complexity Theory program to prove a variant of Valiant's algebraic analog of the P not equal to NP conjecture. We also describe the precise separation of complexity classes that their program proposes to demonstrate.
We prove a general theorem providing smoothed analysis estimates for conic condition numbers of p... more We prove a general theorem providing smoothed analysis estimates for conic condition numbers of problems of numerical analysis. Our probability estimates depend only on geometric invariants of the corresponding sets of ill-posed inputs. Several applications to linear and polynomial equation solving show that the estimates obtained in this way are easy to derive and quite accurate. The main theorem is based on a volume estimate of ε-tubular neighborhoods around a real algebraic subvariety of a sphere, intersected with a disk of radius σ. Besides ε and σ, this bound depends only the dimension of the sphere and on the degree of the defining equations. * Part of these results were announced in C.R. Acad. Sci. Paris, Ser. I 343 (2006) 145-150.
Computing Research Repository, 2010
Mulmuley and Sohoni (GCT1 in SICOMP 2001, GCT2 in SICOMP 2008) proposed to view the permanent ver... more Mulmuley and Sohoni (GCT1 in SICOMP 2001, GCT2 in SICOMP 2008) proposed to view the permanent versus determinant problem as a specific orbit closure problem and to attack it by methods from geometric invariant and representation theory. We adopt these ideas towards the goal of showing lower bounds on the border rank of specific tensors, in particular for matrix multiplication.
Computing Research Repository, 2003
Categories and Subject Descriptors: F.1.1 [Computation by Abstract Devices]: Models of Computatio... more Categories and Subject Descriptors: F.1.1 [Computation by Abstract Devices]: Models of Computation; F.2.1 [Analysis of Algorithms and Problem Complexity]: Numerical Algorithms and Problems—Computations on polynomials; I.1.2 [Symbolic and Algebraic Manipulation]: Algorithms General Terms: Algorithms, Theory Additional Key Words and Phrases: algebraic complexity, bilinear circuits, cyclic convolution, Fast Fourier Transform, lower bounds, polynomial multiplication, singular values, rigidity
The 17th of the problems proposed by Steve Smale for the 21st century asks for the existence of a... more The 17th of the problems proposed by Steve Smale for the 21st century asks for the existence of a deterministic algorithm computing an approximate solution of a system of $n$ complex polynomials in $n$ unknowns in time polynomial, on the average, in the size $N$ of the input system. A partial solution to this problem was given by Carlos Beltran
Corr, Dec 2, 2003
We define counting classes #P_R and #P_C in the Blum-Shub-Smale setting of computations over the ... more We define counting classes #P_R and #P_C in the Blum-Shub-Smale setting of computations over the real or complex numbers, respectively. The problems of counting the number of solutions of systems of polynomial inequalities over R, or of systems of polynomial equalities over C, respectively, turn out to be natural complete problems in these classes. We investigate to what extent the new counting classes capture the complexity of computing basic topological invariants of semialgebraic sets (over R) and algebraic sets (over C). We prove that the problem of computing the (modified) Euler characteristic of semialgebraic sets is FP_R^{#P_R}-complete, and that the problem of computing the geometric degree of complex algebraic sets is FP_C^{#P_C}-complete. We also define new counting complexity classes in the classical Turing model via taking Boolean parts of the classes above, and show that the problems to compute the Euler characteristic and the geometric degree of (semi)algebraic sets given by integer polynomials are complete in these classes. We complement the results in the Turing model by proving, for all k in N, the FPSPACE-hardness of the problem of computing the k-th Betti number of the zet of real zeros of a given integer polynomial. This holds with respect to the singular homology as well as for the Borel-Moore homology.
Journal of Complexity, 1993
We investigate the impact of randomization on the complexity of deciding membership in a (semi-)a... more We investigate the impact of randomization on the complexity of deciding membership in a (semi-)algebraic subset X R m . Examples are exhibited where allowing for a certain error probability in the answer of the algorithms the complexity of decision problems decreases. A randomized ( k ; f=; g)decision tree (k R a sub eld) over m will be de ned as a pair (T; ) where a probability measure on some R n and T is a ( k ; f=; g)-decision tree over m+n. We prove a general lower bound on the average decision complexity for testing membership in an irreducible algebraic subset X R m and apply it to k-generic complete intersection of polynomials of the same degree, extending results in 4, 6]. We also give applications to nongeneric cases, such as graphs of elementary symmetric functions, SL(m; R), and determinant varieties, extending results in Li 90].
SIAM Journal on Computing, 2011
We discuss the geometry of orbit closures and the asymptotic behavior of Kronecker coefficients i... more We discuss the geometry of orbit closures and the asymptotic behavior of Kronecker coefficients in the context of the Geometric Complexity Theory program to prove a variant of Valiant's algebraic analog of the P = N P conjecture. We also describe the precise separation of complexity classes that their program proposes to demonstrate.
Computing Research Repository - CORR, 2010
Mulmuley and Sohoni (GCT1 in SICOMP 2001, GCT2 in SICOMP 2008) proposed to view the permanent ver... more Mulmuley and Sohoni (GCT1 in SICOMP 2001, GCT2 in SICOMP 2008) proposed to view the permanent versus determinant problem as a specific orbit closure problem and to attack it by methods from geometric invariant and representation theory. We adopt these ideas towards the goal of showing lower bounds on the border rank of specific tensors, in particular for matrix multiplication. We thus study specific orbit closure problems for the group $G = GL(W_1)\times GL(W_2)\times GL(W_3)$ acting on the tensor product $W=W_1\otimes W_2\otimes W_3$ of complex finite dimensional vector spaces. Let $G_s = SL(W_1)\times SL(W_2)\times SL(W_3)$. A key idea from GCT2 is that the irreducible $G_s$-representations occurring in the coordinate ring of the $G$-orbit closure of a stable tensor $w\in W$ are exactly those having a nonzero invariant with respect to the stabilizer group of $w$. However, we prove that by considering $G_s$-representations, as suggested in GCT1-2, only trivial lower bounds on borde...
Journal de Mathématiques Pures et Appliquées, 2006
Smoothed analysis of complexity bounds and condition numbers has been done, so far, on a case by ... more Smoothed analysis of complexity bounds and condition numbers has been done, so far, on a case by case basis. In this paper we consider a reasonably large class of condition numbers for problems over the complex numbers and we obtain smoothed analysis estimates for elements in this class depending only on geometric invariants of the corresponding sets of ill-posed inputs. These estimates are for a version of smoothed analysis proposed in this paper which, to the best of our knowledge, appears to be new. Several applications to linear and polynomial equation solving show that estimates obtained in this way are easy to derive and quite accurate.
Foundations of Computational Mathematics, 2007
We continue the study of counting complexity begun in by proving upper and lower bounds on the co... more We continue the study of counting complexity begun in by proving upper and lower bounds on the complexity of computing the Hilbert polynomial of a homogeneous ideal. We show that the problem of computing the Hilbert polynomial of a smooth equidimensional complex projective variety can be reduced in polynomial time to the problem of counting the number of complex common zeros of a finite set of multivariate polynomials. Moreover, we prove that the more general problem of computing the Hilbert polynomial of a homogeneous ideal is polynomial space hard. This implies polynomial space lower bounds for both the problems of computing the rank and the Euler characteristic of cohomology groups of coherent sheaves on projective space, improving the #P-lower bound in Bach [1].
Annals of Mathematics, 2011
The 17th of the problems proposed by Steve Smale for the 21st century asks for the existence of a... more The 17th of the problems proposed by Steve Smale for the 21st century asks for the existence of a deterministic algorithm computing an approximate solution of a system of n complex polynomials in n unknowns in time polynomial, on the average, in the size N of the input system. A partial solution to this problem was given by Carlos Beltrán and Luis Miguel Pardo who exhibited a randomized algorithm doing so. In this paper we further extend this result in several directions. Firstly, we exhibit a linear homotopy algorithm that efficiently implements a non-constructive idea of Mike Shub. This algorithm is then used in a randomized algorithm, call it LV,à la Beltrán-Pardo. Secondly, we perform a smoothed analysis (in the sense of Spielman and Teng) of algorithm LV and prove that its smoothed complexity is polynomial in the input size and σ −1 , where σ controls the size of of the random perturbation of the input systems. Thirdly, we perform a condition-based analysis of LV. That is, we give a bound, for each system f , of the expected running time of LV with input f . In addition to its dependence on N this bound also depends on the condition of f . Fourthly, and to conclude, we return to Smale's 17th problem as originally formulated for deterministic algorithms. We exhibit such an algorithm and show that its average complexity is N O(log log N) . This is nearly a solution to Smale's 17th problem.
We investigate the impact of randomization on the complexity of deciding membership in a (semi-)a... more We investigate the impact of randomization on the complexity of deciding membership in a (semi-)algebraic subset X R m . Examples are exhibited where allowing for a certain error probability in the answer of the algorithms the complexity of decision problems decreases. A randomized ( k ; f=; g)decision tree (k R a sub eld) over m will be de ned as a pair (T; ) where a probability measure on some R n and T is a ( k ; f=; g)-decision tree over m+n. We prove a general lower bound on the average decision complexity for testing membership in an irreducible algebraic subset X R m and apply it to k-generic complete intersection of polynomials of the same degree, extending results in 4, 6]. We also give applications to nongeneric cases, such as graphs of elementary symmetric functions, SL(m; R), and determinant varieties, extending results in Li 90].
Smale's 17th problem asks for an algorithm which finds an approximate zero of polynomial systems ... more Smale's 17th problem asks for an algorithm which finds an approximate zero of polynomial systems in average polynomial time (see [21]). The main progress on Smale's problem is [6] and [10]. In this paper we will improve on both approaches and prove an interesting intermediate result on the 1 average value of the condition number. Our main results are Theorem 1 on the complexity of a randomized algorithm which improves the result of [6], Theorem 2 on the average of the condition number of polynomial systems which improves the estimate found in [10], and Theorem 3 on the complexity of finding a single zero of polynomial systems. This last theorem is similar to the main result of [10] but relies only on homotopy methods, thus removing the need for the elimination theory methods used in [10]. We build on methods developed in [2].
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Papers by Peter Buergisser