# On the additive complexity of polynomials and some new lower bounds

• C. P. Schnorr
Vorträge (In Alphabetischer Reihenfolge)
Part of the Lecture Notes in Computer Science book series (LNCS, volume 67)

## Abstract

For each w ∈ N we establish polynomials Rw,j j ∈ N with (w+1) (w+2) / 2 variables and degRw,j≤2wj+1 such that the coefficient vectors (aj | j ∈ N) of all polynomials Σjaj(x-η)j which can be computed with ≤w additions/subtractions and arbitrarily many mult./div., are contained in the image of (Rw+1,j | j ∈ N). As a consequence we prove c 0,1 t (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 0,1 t (n), c 0,1 ns (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 $$L_{ns} \left( {\sum _{i = 1}^k x_i^n y^i } \right) \gtrsim {{k ld n} \mathord{\left/{\vphantom {{k ld n} {(ld k + ld ld n)}}} \right.\kern-\nulldelimiterspace} {(ld k + ld ld n)}}$$.

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