On the additive complexity of polynomials and some new lower bounds

Abstract

For each w ∈ N we establish polynomials R_w,j j ∈ N with (w+1) (w+2) / 2 variables and degR_w,j≤2wj+1 such that the coefficient vectors (a_j | j ∈ N) of all polynomials Σ_ja_j(x-η)^j which can be computed with ≤w additions/subtractions and arbitrarily many mult./div., are contained in the image of (R_w+1,j | j ∈ N). As a consequence we prove c ^t_0,1 (n)≥n/ (8ld(n)+4)−1 (this bound is sharp up to a constant factor), $c_{O, 1}^{ns} \left( n \right) \geqslant \tfrac{1}{4}\sqrt {{n \mathord{\left/{\vphantom {n {(ld(2n))}}} \right.\kern-\nulldelimiterspace} {(ld(2n))}}} - 2$ and $c_{O, 1}^ + \left( n \right) \geqslant {{\sqrt n } \mathord{\left/{\vphantom {{\sqrt n } {\left( {4ld n} \right)}}} \right.\kern-\nulldelimiterspace} {\left( {4ld n} \right)}}$. Hereby c ^t_0,1 (n), c ^ns_0,1 (n) and cc ⁺₀ (n) are the maximal number of arithmetical operations, non-scalar operations and add./sub. respectively that are necessary to evaluate n degree polynomials with 0–1 coefficients. We specify n-degree polynomials with algebraic coefficients that require n additions/subtractions no matter how many mult./div. are used. As a first non-trivial lower bound on a single specific polynomial with integer coefficients we prove $L_{ns} \left( {\sum _{i = 1}^k x_i^n y^i } \right) \gtrsim {{k ld n} \mathord{\left/{\vphantom {{k ld n} {(ld k + ld ld n)}}} \right.\kern-\nulldelimiterspace} {(ld k + ld ld n)}}$.