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We prove exponential lower bounds on the size of homogeneous depth 4 arithmetic circuits computing an explicit polynomial in VP. Our results hold for the Iterated Matrix Multiplication polynomial - in particular we show that any homogeneous depth 4 circuit computing the (1, 1) entry in the product of n generic matrices of dimension n<sup>O(1)</sup> must(More)
We introduce the polynomial coefficient matrix and identify maximum rank of this matrix under variable substitution as a complexity measure for multivariate polyno-mials. We use our techniques to prove super-polynomial lower bounds against several classes of non-multilinear arithmetic circuits. In particular, we obtain the following results : • As our main(More)
An undirected graph is said to be split if its vertex set can be partitioned into two sets such that the subgraph induced on one of them is a complete graph and the subgraph induced on the other is an independent set. We initiate a systematic study of parameterized complexity of the problem of deleting the minimum number of vertices or edges from a given(More)
In this paper, we prove superpolynomial lower bounds for the class of homogeneous depth 4 arithmetic circuits. We give an explicit polynomial in VNP of degree n in n 2 variables such that any homogeneous depth 4 arithmetic circuit computing it must have size n Ω(log log n). Our results extend the works of Nisan-Wigderson [NW95] (which showed superpolynomial(More)
We say that a circuit C over a field F functionally computes an n-variate polynomial P ∈ F[x 1 , x 2 ,. .. , x n ] if for every x ∈ {0, 1} n we have that C(x) = P(x). This is in contrast to syntactically computing P, when C ≡ P as formal polynomials. In this paper, we study the question of proving lower bounds for homogeneous depth-3 and depth-4 arithmetic(More)
In recent years, a very exciting and promising method for proving lower bounds for arithmetic circuits has been proposed. This method combines the method of <i>depth reduction</i> developed in the works of Agrawal and Vinay[1], Koiran [11] and Tavenas [16], and the use of the shifted partial derivative complexity measure developed in the works of Kayal [9](More)
We study the complexity of representing polynomials as a sum of products of polynomials in few variables. More precisely, we study representations of the form P = T i=1 d j=1 Q ij such that each Q ij is an arbitrary polynomial that depends on at most s variables. We prove the following results. Over fields of characteristic zero, for every constant µ such(More)
Piwi-interacting RNAs (piRNAs) and CRISPR RNAs (crRNAs) are two recently discovered classes of small noncoding RNA that are found in animals and prokaryotes, respectively. Both of these novel RNA species function as components of adaptive immune systems that protect their hosts from foreign nucleic acids-piRNAs repress transposable elements in animal(More)
In recent years there has been a flurry of activity proving lower bounds for homogeneous depth-4 arithmetic circuits [GKKS13, FLMS14, KLSS14, KS14c], which has brought us very close to statements that are known to imply VP = VNP. It is a big question to go beyond homogeneity, and in this paper we make progress towards this by considering depth-4 circuits of(More)
In this paper, we show exponential lower bounds for the class of homogeneous depth-5 circuits over all small finite fields. More formally, we show that there is an explicit family {P d : d ∈ N} of polynomials in VNP, where P d is of degree d in n = d O(1) variables, such that over all finite fields F q , any homogeneous depth-5 circuit which computes P d(More)