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Polynomial identity testing for depth 3 circuits
 in Proceedings of the twentyfirst Annual IEEE Conference on Computational Complexity (CCC
, 2006
"... Abstract — We study ΣΠΣ(k) circuits, i.e., depth three arithmetic circuits with top fanin k. We give the first deterministic polynomial time blackbox identity test for ΣΠΣ(k) circuits over the field Q of rational numbers, thus resolving a question posed by Klivans and Spielman (STOC 2001). Our main ..."
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Abstract — We study ΣΠΣ(k) circuits, i.e., depth three arithmetic circuits with top fanin k. We give the first deterministic polynomial time blackbox identity test for ΣΠΣ(k) circuits over the field Q of rational numbers, thus resolving a question posed by Klivans and Spielman (STOC 2001). Our main technical result is a structural theorem for ΣΠΣ(k) circuits that compute the zero polynomial. In particular we show that if a ΣΠΣ(k) circuit C = ∑ i∈[k] Ai
FROM SYLVESTERGALLAI CONFIGURATIONS TO RANK BOUNDS: IMPROVED BLACKBOX IDENTITY TEST FOR DEPTH3 CIRCUITS
"... Abstract. We study the problem of identity testing for depth3 circuits of top fanin k and degree d (called ΣΠΣ(k, d) identities). We give a new structure theorem for such identities. A direct application of our theorem improves the known deterministic d kO(k) time blackbox identity test over ratio ..."
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Cited by 13 (2 self)
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Abstract. We study the problem of identity testing for depth3 circuits of top fanin k and degree d (called ΣΠΣ(k, d) identities). We give a new structure theorem for such identities. A direct application of our theorem improves the known deterministic d kO(k) time blackbox identity test over rationals (Kayal & Saraf, FOCS 2009) to one that takes d O(k2)time. Our structure theorem essentially says that the number of independent variables in a real depth3 identity is very small. This theorem settles affirmatively the stronger rank conjectures posed by Dvir & Shpilka (STOC 2005) and Kayal & Saraf (FOCS 2009). Our techniques provide a unified framework that actually beats all known rank bounds and hence gives the best running time (for every field) for blackbox identity tests. Our main theorem (almost optimally) pins down the relation between higher dimensional SylvesterGallai theorems and the rank of depth3 identities in a very transparent manner. The existence of this was hinted at by Dvir & Shpilka (STOC 2005), but first proven, for reals, by Kayal & Saraf (FOCS 2009). We introduce the concept of SylvesterGallai rank bounds for any field, and show the intimate connection between this and depth3 identity rank bounds. We also prove the first ever theorem about high dimensional SylvesterGallai configurations over any field. Our proofs and techniques are very different from previous results and devise a very interesting ensemble of combinatorics and algebra. The latter concepts are ideal theoretic and involve a new Chinese remainder theorem. Our proof methods explain the structure of any depth3 identity C: there is a nucleus of C that forms a low rank identity, while the remainder is a high dimensional SylvesterGallai configuration. 1.
Rank bounds for design matrices with applications to combinatorial geometry and locally correctable codes
 Proc. of the 43rd annual STOC, ACM Press
, 2011
"... A (q, k, t)design matrix is an m × n matrix whose pattern of zeros/nonzeros satisfies the following designlike condition: each row has at most q nonzeros, each column has at least k nonzeros and the supports of every two columns intersect in at most t rows. We prove that for m ≥ n, the rank of ..."
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Cited by 7 (4 self)
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A (q, k, t)design matrix is an m × n matrix whose pattern of zeros/nonzeros satisfies the following designlike condition: each row has at most q nonzeros, each column has at least k nonzeros and the supports of every two columns intersect in at most t rows. We prove that for m ≥ n, the rank of any (q, k, t)design matrix over a field of characteristic zero (or sufficiently large finite characteristic) is at least n − ( ) 2 qtn 2k Using this result we derive the following applications: Impossibility results for 2query LCCs over large fields. A 2query locally correctable code (LCC) is an error correcting code in which every codeword coordinate can be recovered, probabilistically, by reading at most two other code positions. Such codes have numerous applications and constructions (with exponential encoding length) are
On the number of directions determined by a threedimensional points set
 J. Combin. Theory Ser. A
"... Let P be a set of n points in R 3, not all of which are in a plane and no three on a line. We partially answer a question of Scott (1970) by showing that the connecting lines of P assume at least 2n − 3 different directions if n is even and at least 2n − 2 if n is odd. These bounds are sharp. The pr ..."
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Cited by 7 (2 self)
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Let P be a set of n points in R 3, not all of which are in a plane and no three on a line. We partially answer a question of Scott (1970) by showing that the connecting lines of P assume at least 2n − 3 different directions if n is even and at least 2n − 2 if n is odd. These bounds are sharp. The proof is based on a farreaching generalization of Ungar’s theorem concerning the analogous problem in the plane. 1
Progress on Polynomial Identity Testing
"... Polynomial identity testing (PIT) is the problem of checking whether a given arithmetic circuit is the zero circuit. PIT ranks as one of the most important open problems in the intersection of algebra and computational complexity. In the last few years, there has been an impressive progress on this ..."
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Polynomial identity testing (PIT) is the problem of checking whether a given arithmetic circuit is the zero circuit. PIT ranks as one of the most important open problems in the intersection of algebra and computational complexity. In the last few years, there has been an impressive progress on this problem but a complete solution might take a while. In this article we give a soft survey exhibiting the ideas that have been useful. 1
An algorithmic proof of the MotzkinRabin theorem on monochrome lines
, 2002
"... We present a new proof of the following theorem originally due to Motzkin and Rabin (see [9, 1, 6, 7]). MotzkinRabin Theorem. Let S be a finite noncollinear set of points in the plane, each colored red or blue. Then there exists a line l passing through at least two points of S, all points of S on ..."
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Cited by 5 (1 self)
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We present a new proof of the following theorem originally due to Motzkin and Rabin (see [9, 1, 6, 7]). MotzkinRabin Theorem. Let S be a finite noncollinear set of points in the plane, each colored red or blue. Then there exists a line l passing through at least two points of S, all points of S on l being of the same color. We say that a set of points S is twocolored if each point in S is assigned one of the colors red or blue. A line passing through at least two points of S with all points of S on the line assigned the same color is called a monochrome line. It makes no di erence whether the plane in the theorem is the Euclidean or the projective plane, since if we are in the projective plane we can always find a line disjoint from the finite set S, project it to infinity, and then the set S can be considered to be in the Euclidean plane. There are two essentially different proofs of the MotzkinRabin theorem in the literature, both proving the projective dual of the theore...
The SylvesterChvatal Theorem
 Discrete & Computational Geometry
"... The SylvesterGallai theorem asserts that every finite set S of points in twodimensional Euclidean space includes two points, a and b, such that either there is no other point in S is on the line ab, or the line ab contains all the points in S.V.Chvatal extended the notion of lines to arbitrary ..."
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Cited by 4 (2 self)
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The SylvesterGallai theorem asserts that every finite set S of points in twodimensional Euclidean space includes two points, a and b, such that either there is no other point in S is on the line ab, or the line ab contains all the points in S.V.Chvatal extended the notion of lines to arbitrary metric spaces and made a conjecture that generalizes the SylvesterGallai theorem. In the present article we prove this conjecture.
SylvesterGallai theorem and metric betweenness
, 2002
"... Sylvester conjectured in 1893 and Gallai proved some forty years later that every finite set S of points in the plane includes two points such that the line passing through them includes either no other point of S or all other points of S. There are several ways of extending the notion of lines fro ..."
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Cited by 4 (0 self)
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Sylvester conjectured in 1893 and Gallai proved some forty years later that every finite set S of points in the plane includes two points such that the line passing through them includes either no other point of S or all other points of S. There are several ways of extending the notion of lines from Euclidean spaces to arbitrary metric spaces. We present one of them and conjecture that, with lines in metric spaces defined in this way, the SylvesterGallai theorem generalizes as follows: in every finite metric space, there is a line consisting of either two points or all the points of the space. Then we present slight evidence in support of this rash conjecture and finally we discuss the underlying ternary relation of metric betweenness. 1 The SylvesterGallai theorem Sylvester (1893) proposed the following problem: Prove that it is not possible to arrange any finite number of real points so that a right line through every two of them shall pass through a third, unless they all line in the same right line.
THE SYLVESTERGALLAI THEOREM, COLOURINGS AND ALGEBRA
, 2006
"... Our point of departure is the following simple common generalisation of the SylvesterGallai theorem and the MotzkinRabin theorem: Let S be a finite set of points in the plane, with each point coloured red or blue or with both colours. Suppose that for any two distinct points A, B ∈ S sharing a co ..."
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Cited by 3 (0 self)
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Our point of departure is the following simple common generalisation of the SylvesterGallai theorem and the MotzkinRabin theorem: Let S be a finite set of points in the plane, with each point coloured red or blue or with both colours. Suppose that for any two distinct points A, B ∈ S sharing a colour there is a third point C ∈ S, of the other colour, collinear with A and B. Then all the points in S are collinear. We define a chromatic geometry to be a simple matroid for which each point is coloured red or blue or with both colours, such that for any two distinct points A, B ∈ S sharing a colour there is a third point C ∈ S, of the other colour, collinear with A and B. This is a common generalisation of proper finite linear spaces and properly twocoloured finite linear spaces, with many known properties of both generalising as well. One such property is Kelly’s complex SylvesterGallai theorem. We also consider embeddings of chromatic geometries in Desarguesian projective spaces. We prove a lower bound of 51 for the number of points in a 3dimensional chromatic geometry in projective space over the quaternions. Finally, we suggest an elementary approach to the corollary of an inequality of Hirzebruch used by Kelly in his proof of the complex SylvesterGallai theorem.