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.

finite fields*

We perform forward error analysis for a large class of recursive matrix multiplication algorithms in the spirit of [D. Bini and G. Lotti, Stability of fast algorithms for matrix multiplication, Numer. Math. 36 (1980), 63–72]. As a consequence of our analysis, we show that the exponent of matrix multiplication can be achieved by numerically stable algorithms. We also show that new group-theoretic algorithms proposed in [H. Cohn, and C. Umans, A group-theoretic approach to fast matrix multiplication, FOCS 2003, 438–449] and [H. Cohn, R. Kleinberg, B. Szegedy and C. Umans, Group-theoretic algorithms for matrix multiplication, FOCS 2005, 379–388] are all included in the class of algorithms to which our analysis applies, and are therefore numerically stable. We perform detailed error analysis for three specific fast group-theoretic algorithms. 1

Let R(A) denote the rank (also called bilinear complexity) of a finite dimensional associative algebra A. A fundamental lower bound for R(A) is the so-called Alder-- Strassen bound R(A) 2 dim A \Gamma t, where t is the number of maximal twosided ideals of A. The class of algebras for which the Alder--Strassen bound is sharp, the so-called algebras of minimal rank, has received a wide attention in algebraic complexity theory.

Linear spaces of n \Theta n \Theta n tensors over finite fields are investigated where the rank of every nonzero tensor in the space is bounded from below by a prescribed number ¯. Such linear paces can recover any n \Theta n \Theta n error tensor of rank (¯\Gamma1)=2, and, as such, they can be used to correct three-way crisscross errors. Bounds on the dimensions of such spaces are given for ¯ 2n+1, and constructions are provided for ¯ 2n\Gamma1 with redundancy which is linear in n. These constructions can be generalized to spaces of n \Theta n \Theta \Delta \Delta \Delta \Theta n hyper-arrays. Keywords: Algebraic computation, Crisscross errors, Tensor rank. This work was presented in part at the IEEE International Symposium on Information Theory, Whistler, BC, Canada, September 1995. y This research was supported by the Fund for the Promotion of Research at the Technion and by the Technion V.P.R Steiner Research Fund. 1 Introduction An n \Theta n \Theta n tensor over a fiel...

. We prove the 2.5n - o (n ) lower bound on the number of multiplications/divisions required to compute the coefficients of the product of two polynomials of degree n over a finite field by means of straight-line algorithms. 1. Introduction The number of multiplications/divisions required for computing the product of two degree-n polynomials over an infinite field is known to be 2n + 1. The method is to evaluate both polynomials at each of 2n + 1 distinct points (allowing ), multiplying and interpolating the result. This method fails for the fields with the number of elements less than 2n . The bilinear and quadratic complexity of polynomial multiplication over finite fields has been widely studied in the literature, cf. [BD1], [BD2], [J2], [KA], [KB2], [LSW] and [LW]. The best known lower bounds on the bilinear complexity of multiplying two degree-n polynomials over finite fields are as follows. For the binary field the bound is 3.52n , cf. [BD2]. The same bound holds for the quadrati...

It is shown that the maximal rank of m × n × ( m n - k ) tensors with k min {( m - 1 ) 2 /2 , ( n - 1 ) 2 /2} is greater than m n - 4Ö ### 2 k + O ( 1 ) . 1. INTRODUCTION A classical problem in algebraic computational complexity is to determine the minimal number of non-scalar multiplications required to evaluate some set S i ,j a i ,j ,k x i y j , k = 1, . . . , p , of bilinear forms in noncommuting variables x 1 , . . . , x m and y 1 , . . . , y n over a field F . This number is equal to the rank of the defining 3-tensor ( a i ,j ,k ) ÎF m ÄF n ÄF p , cf. [S]. An interesting problem, which does not depend on the coefficients a i ,j ,k , is the determination of R F ( m , n , p ) = max T ÎF m ÄF n ÄF p rank of T , the maximal rank of tensors in F m ÄF n ÄF p . This problem has been studied quite extensively in [AL1AS, Gat-J]. Atkinson and Stephens [AS] gave the following general reduction of R F ( m , n , m n - k ) with k min { m , n } to R F ( k , k , k 2 ...

We prove a lower bound of 5/2 n² - 3n for the rank of n×n-matrix multiplication over an arbitrary field. Similar bounds hold for the rank of the multiplication in noncommutative division algebras and for the multiplication of upper triangular matrices.

We prove a lower bound of \Gamma 3n for the multiplicative complexity of n × n-matrix multiplication over arbitrary fields. More general, we show that for any finite dimensional semisimple algebra A with unity, the multiplicative complexity of the multiplication in A is bounded from below by 2 dimA \Gamma 3(n1 + \Delta \Delta \Delta + n t ) if the decomposition of A = A1 \Theta \Delta \Delta \Delta \Theta A t into simple algebras A = D contains only noncommutative factors, that is, the division algebra D is noncommutative or n ≥ 2.

We prove a lower bound of � Ò   Ò for the rank of Ò ¢ Ò–matrix multiplication over an arbitrary field. Similar bounds hold for the rank of the multiplication in noncommutative division algebras and for the multiplication of upper triangular matrices.