Elusive Functions And Lower Bounds For Arithmetic Circuits

A basic fact in linear algebra is that the image of the curve f (x) = (x(1), x(2), x(,)(3) ... ,x(m)), say over C, is not contained in any m - 1. dimensional affine subspace of C-m. In other words, the image of f is not contained in the image of any polynomial-mapping(1) Gamma : Cm-1 --> C-m of degree 1 (that is, air affine mapping). Can one give an explicit example for a polynomial curve f : C --> C-m, such that, the image of f is not contained in the image of any polynomial-mapping Gamma : Cm-1 --> C-m of degree 2 ?In this paper, we show that problems of this type are closely related to proving lower bounds for the size of general arithmetic circuits. [For example, any explicit f as above (with the right notion of explicitness 2), of degree up to 2(m degrees(1)), implies super-polynomial lower bounds for computing the permanent over C.More generally, we say that a polynomial-mapping f F-n --> F-m is (s,r)-elusive, if for every polynomial-mapping Gamma : F-s --> F-m of degree r, Image(f) not subset of Image(Gamma). We show that for many settings of the parameters explicit constructions of elusive polynomial-mappings imply strong (up to exponential) lower bounds for general arithmetic circuits.Finally, for every r < log n, we give art explicit example for a polynomial-mapping f : F-n --> F-n2, of degree O(r), that is (s,r)-elusive for s = n(1+Omega(1/r)). We use this to construct for any r, air explicit example for art n-variate polynomial of total-degree O(r), with coefficients in {0,1}, such that, any depth r arithmetic circuit for this polynomial (over any field) is of size >= n(1+Omega(1/r)).In particular, for any constant r, this gives a constant degree polynomial, such that, any depth r arithmetic circuit for this polynomial is of. size >= n(1+Omega(1)). Previously, only lower bounds of the type Omega(n . lambda(r)(n)), where lambda(r)(n) are extremely slowly growing functions (e.g., lambda(5)(n) = log* n, and lambda(7)(n) = log* log* n), were known for constant-depth arithmetic circuits for polynomials of constant degree.