Constructing elusive functions with help of evaluation mappings

We introduce a general concept of complexity and a polynomial type algebraic complexity of a polynomial mapping. The notion of polynomial type algebraic complexity encompasses the determinantal complexity. We analyze the relation between polynomial type algebraic complexities and elusive functions. We study geometric-algebraic properties of polynomial type computational complexities. We present two algorithms to construct a test function for estimating a polynomial type algebraic complexity; one of them is based on Gröbner bases, the other uses the resultant of polynomials in many variables. We describe an algorithm to compute a polynomial type algebraic complexity of polynomial mappings defined over C. We develop the method by KumarLokam-Patankar-Sarma that uses the effective elimination theory combined with algebraic number field theory in order to construct elusive functions and polynomial mappings with large circuit size. For F = C or R, and for any given r, we construct explicit examples of sequences of polynomial mappings fn : F 2n → Fn and of degree 5r+1 whose coefficients are algebraic numbers such that any depth r arithmetic circuit for fn is of size greater than n /(50r). Using the developed methods we also construct concrete examples of polynomials with large determinantal complexity. AMSC: 03D15, 68Q17, 13P25