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Sparse shifts for univariate polynomials

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Abstract

Letf(x) be a polynomial of degreed with rational coefficients and lett be a positive integer ⩽ deg(f). We consider the problem of finding at-sparse shift forf(x). The problem is to find an a, if one exists (in some algebraic extension of the rationals), such that in the representation off(x) in the basis 1,x − α, (x − α)2,..., i.e.,\(f(x) = \sum\nolimits_{i = 0^{F_i } }^d {(x - \alpha )^i } \) at most t of the coefficients fi are non-zero. We derive explicit conditions for the uniqueness and rationality of at-sparse shift forf(x) and provide an efficient algorithm for computing a sparse shift when one exists. We also point out an application of our result to the problem of constructing sparse decompositions of univariate polynomials.

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Authors and Affiliations

  1. Department of Mathematics and Computer Science, Drexel University, 19104, Philadelphia, PA, USA

    Y. N. Lakshman

  2. Department of Computer and Information Sciences, University of Delaware, 19716, Newark, DE, USA

    B. David Saunders

Authors
  1. Y. N. Lakshman
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  2. B. David Saunders
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Additional information

Work by Y. N. Lakshman was supported by NSF grant CCR-9203062

Work by B. D. Saunders was supported by NSF grant CCR-9123666

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Lakshman, Y.N., Saunders, B.D. Sparse shifts for univariate polynomials. AAECC 7, 351–364 (1996). https://doi.org/10.1007/BF01293594

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  • DOI: https://doi.org/10.1007/BF01293594

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