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Geometry and the complexity of matrix multiplication
, 2007
"... Abstract. We survey results in algebraic complexity theory, focusing on matrix multiplication. Our goals are (i) to show how open questions in algebraic complexity theory are naturally posed as questions in geometry and representation theory, (ii) to motivate researchers to work on these questions, ..."
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Cited by 15 (1 self)
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Abstract. We survey results in algebraic complexity theory, focusing on matrix multiplication. Our goals are (i) to show how open questions in algebraic complexity theory are naturally posed as questions in geometry and representation theory, (ii) to motivate researchers to work on these questions, and (iii) to point out relations with more general problems in geometry. The key geometric objects for our study are the secant varieties of Segre varieties. We explain how these varieties are also useful for algebraic statistics, the study of phylogenetic invariants, and quantum computing.
Generalising the HardyLittlewood method for primes
 In: Proceedings of the international congress of mathematicians
, 2007
"... Abstract. The HardyLittlewood method is a wellknown technique in analytic number theory. Among its spectacular applications are Vinogradov’s 1937 result that every sufficiently large odd number is a sum of three primes, and a related result of Chowla and Van der Corput giving an asymptotic for the ..."
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Cited by 5 (2 self)
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Abstract. The HardyLittlewood method is a wellknown technique in analytic number theory. Among its spectacular applications are Vinogradov’s 1937 result that every sufficiently large odd number is a sum of three primes, and a related result of Chowla and Van der Corput giving an asymptotic for the number of 3term progressions of primes, all less than N. This article surveys recent developments of the author and T. Tao, in which the HardyLittlewood method has been generalised to obtain, for example, an asymptotic for the number of 4term arithmetic progressions of primes less than N.
WARING’S PROBLEM RESTRICTED BY A SYSTEM OF SUM OF DIGITS CONGRUENCES
, 2007
"... Abstract. The aim of the present paper is to generalize earlier work by Thuswaldner and Tichy on Waring’s Problem with digital restrictions to systems of digital restrictions. Let sq(n) be the qadic sum of digits function and let d, s, al, ml, ql ∈ N. Then for s> d 2 ( log d + log log d + O(1) ) th ..."
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Cited by 2 (0 self)
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Abstract. The aim of the present paper is to generalize earlier work by Thuswaldner and Tichy on Waring’s Problem with digital restrictions to systems of digital restrictions. Let sq(n) be the qadic sum of digits function and let d, s, al, ml, ql ∈ N. Then for s> d 2 ( log d + log log d + O(1) ) there exists N0 ∈ N such that each integer N ≥ N0 has a representation of the form N = x d 1 + · · · + x d s where sql (xi) ≡ al mod ml (1 ≤ i ≤ s and 1 ≤ l ≤ L). The result, together with an asymptotic formula of the number of this representations, will be shown with the help of the circle method together with exponential sum estimates. 1. Notation Let N, Z and R denote the set of positive integers, integers and real numbers, respectively. A set of the shape {n ∈ Z  a ≤ n ≤ b} will be called interval of integers. The notations e(z) for exp(2πiz), ⌊x ⌋ for the greatest integer less than or equal to x ∈ R, and ⌈x ⌉ for the smallest integer greater than or equal to x will be used frequently. For the sake of shortness, we are going to make extensive use of vector and matrix notation throughout this paper. For example, if v1,..., vd is a finite collection of indexed numbers, then v = (v1,...,vd) will denote the corresponding vector. Furthermore we will use the notations f(x) = O ( g(x) ) as well as f(x) ≪ g(x) to express that f(x)  ≤ cg(x)  for some positive constant c and all sufficiently large x ∈ R. A function f is said to be completely qadditive, if for any p, r, t ∈ N with 0 ≤ r < q t the property f(p ·q t +r) = f(p)+f(r) holds. The classical example of a completely qadditive function is the the qadic sum of digits function sq which assigns to each positive integer n the sum sq(n) = c0 + · · · + cr of digits in its (unique) qadic representation n = c0 + c1q + · · · + crq r. This function will play a prominent role throughout the paper.
Author manuscript, published in "Mathematische Zeitschrift (2011)" DOI: 10.1007/s0020901109076 DECOMPOSITION OF HOMOGENEOUS POLYNOMIALS WITH LOW RANK
, 2011
"... ABSTRACT: Let F be a homogeneous polynomial of degree d in m + 1 variables defined over an algebraically closed field of characteristic 0 and suppose that F belongs to the sth secant variety of the duple Veronese embedding of Pm into P (m+d d)−1 but that its minimal decomposition as a sum of dth ..."
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ABSTRACT: Let F be a homogeneous polynomial of degree d in m + 1 variables defined over an algebraically closed field of characteristic 0 and suppose that F belongs to the sth secant variety of the duple Veronese embedding of Pm into P (m+d d)−1 but that its minimal decomposition as a sum of dth powers of linear forms M1,..., Mr is F = M d 1 + · · ·+M d r with r> s. We show that if s+r ≤ 2d+1 then such a decomposition of F can be split in two parts: one of them is made by linear forms that can be written using only two variables, the other part is uniquely determined once one has fixed the first part. We also obtain a uniqueness theorem for the minimal decomposition of F if r is at most d and a mild condition is satisfied.
Decomposition of homogeneous polynomials with low rank
"... Abstract Let F be a homogeneous polynomial of degree d in m + 1 variables defined over an algebraically closed field of characteristic 0 and suppose that F belongs to the sth secant variety of the duple Veronese embedding of Pm () m + d d −1 into P but that its minimal decomposition as a sum of d ..."
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Abstract Let F be a homogeneous polynomial of degree d in m + 1 variables defined over an algebraically closed field of characteristic 0 and suppose that F belongs to the sth secant variety of the duple Veronese embedding of Pm () m + d d −1 into P but that its minimal decomposition as a sum of dth powers of linear forms M1,...,Mr is F = Md 1 +···+Md r with r> s. We show that if s +r ≤ 2d +1 then such a decomposition of F can be split in two parts: one of them is made by linear forms that can be written using only two variables, the other part is uniquely determined once one has fixed the first part. We also obtain a uniqueness theorem for the minimal decomposition of F if r is at most d and a mild condition is satisfied.
1.1.2 Exponential Sums........................ 8
"... In this thesis we present an adaptation of the HardyLittlewood Circle Method to give estimates for the number of curves in a variety over a finite field. The key step in the classical Circle Method is to prove that some cancellation occurs in some exponential sums. Using a theorem of Katz, we reduc ..."
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In this thesis we present an adaptation of the HardyLittlewood Circle Method to give estimates for the number of curves in a variety over a finite field. The key step in the classical Circle Method is to prove that some cancellation occurs in some exponential sums. Using a theorem of Katz, we reduce this to bounding the dimension of some singular loci. The method is fully carried out to estimate the number of rational curves in a Fermat hypersurface of low degree and some suggestions are given as to how to handle other cases. We draw geometrical consequences from the main estimates, for instance the irreducibility of the space of rational curves on a Fermat hypersurface in a given degree range, and a bound on the dimension of the singular locus of the moduli