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 t0,1 (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 t0,1 (n), c ns0,1 (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 \(L_{ns} \left( {\sum _{i = 1}^k x_i^n y^i } \right) \gtrsim {{k ld n} \mathord{\left/{\vphantom {{k ld n} {(ld k + ld ld n)}}} \right.\kern-\nulldelimiterspace} {(ld k + ld ld n)}}\).
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References
Belaga, E.G.: (1958) Some problems involved in the computation of polynomials. Dokl. Akad. Nauk. 123, 775–777
Borodin, A. and Cook, S.: (1976) On the number of additions to compute specific polynomials, Siam J. Comput. 5, 146–157
Borodin, A. and Munro, I.: (1975) The complexity of algebraic and numeric problems. American Elsevier, New York
Heintz, J.: (1978) A new method for proving lower bounds for polynomials which are hard to compute. This symposium
Hyafil, L. and Van de Wiele, J.P.: (1976) Bornes Inférieures pour la complexité des polynomes à coefficients 0–1. IRIA Rapport No. 192
Lipton, R.J.: (1975) Polynomials with 0–1 coefficients that are hard to compute. in: Proceedings of the 16th Annual IEEE Symposium on the Foundations of Computer Science, New York
Lipton, R.J. and Stockmeyer, L.J.: (1978) Evaluation of polynomials with super-preconditioning. Journal of Comp. and System Sciences 16, 124–139
Motzkin, T.S.: (1955) Evaluation of polynomials and evaluation of rational functions Bull. Amer. Math. Soc. 61, 163
Paterson, M.S. and Stockmeyer, L.J.: (1973) On the number of non-scalar multiplications necessary to evaluate polynomials. Siam J. Comput. 2, 60–66
Savage, J.E.: (1974) An algorithm for the computation of linear forms. Siam J. Comput. 3, 150–158
Schnorr, C.P.: (1977) Improved lower bounds on the number of multiplications/divisions which are necessary to evaluate polynomials. in: Proceedings of the 6th International MFCS Symposium, High Tatras. Springer: Lecture Notes in Computer Science 53, 135–147. to appear in TCS (1978)
Schnorr, C.P.: (1978) On the additive complexity of polynomials. preprint Universität Frankfurt
Strassen, V.: (1974) Polynomials with rational coefficients which are hard to compute. Siam J. Comput. 3, 128–149
Van de Wiele, J.P.: (1978) An optimal lower bound on the number of total operations to compute 0–1 polynomials over the field of complex numbers. Proceedings of the 19th Annual Symposium on Foundations of Computer Science
Winograd, S.: (1970) On the number of multiplications necessary to compute certain functions. Comm. Pure Appl. Math. 23, 165–179
Strassen, V.: (1973) Die Berechnungskomplexität von elementarsymmetrischen Funktionen und von Interpolationskoeffizienten. Numerische Mathematik 20, 238–251
Schnorr, C.P. and Van de Wiele, J.P.: On the additive complexity of polynomials. To appear in TCS.
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Schnorr, C.P. (1979). On the additive complexity of polynomials and some new lower bounds. In: Weihrauch, K. (eds) Theoretical Computer Science 4th GI Conference. Lecture Notes in Computer Science, vol 67. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-09118-1_30
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DOI: https://doi.org/10.1007/3-540-09118-1_30
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