### Tensor-rank and lower bounds for arithmetic formulas

Ran Raz
2010 Proceedings of the 42nd ACM symposium on Theory of computing - STOC '10
We show that any explicit example for a tensor A : [n] r → F with tensor-rank ≥ n r·(1−o(1)) , where r = r(n) ≤ log n/ log log n is super-constant, implies an explicit super-polynomial lower bound for the size of general arithmetic formulas over F. This shows that strong enough lower bounds for the size of arithmetic formulas of depth 3 imply super-polynomial lower bounds for the size of general arithmetic formulas. One component of our proof is a new approach for homogenization and
more » ... zation of arithmetic formulas, that gives the following results: We show that for any n-variate homogeneous polynomial f of degree r, if there exists a (fanin-2) formula of size s and depth d for f then there exists a homogeneous formula of size O d+r+1 r · s for f . In particular, for any r ≤ O(log n), if there exists a polynomial size formula for f then there exists a polynomial size homogeneous formula for f . This refutes a conjecture of Nisan and Wigderson [NW95] and shows that super-polynomial lower bounds for homogeneous formulas for polynomials of small degree imply super-polynomial lower bounds for general formulas. We show that for any n-variate set-multilinear polynomial f of degree r, if there exists a (fanin-2) formula of size s and depth d for f then there exists a set-multilinear formula of size O ((d + 2) r · s) for f . In particular, for any r ≤ O(log n/ log log n), if there exists a polynomial size formula for f then there exists a polynomial size set-multilinear formula for f . This shows that super-polynomial lower bounds for set-multilinear formulas for polynomials of small degree imply super-polynomial lower bounds for general formulas.