Sparse polynomial

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In mathematics, a sparse polynomial is a polynomial that has far fewer terms than its degree and number of variables would suggest. Examples include

  • monomials, polynomials with only one term,
  • binomials, polynomials with only two terms, and
  • trinomials, polynomials with only three terms.

Sparse polynomials have also been called lacunary polynomials[1] and fewnomials.[2]

Research on sparse polynomials has included work on algorithms whose running time grows as a function of the number of terms rather than on the degree,[3] for problems including polynomial multiplication,[4] root-finding algorithms,[5] and polynomial greatest common divisors.[6] Sparse polynomials have also been used in pure mathematics, especially in the study of Galois groups, because it has been easier to determine the Galois groups of certain families of sparse polynomials than it is for other polynomials.[7] The algebraic varieties determined by sparse polynomials have a simple structure, which is also reflected in the structure of the solutions of certain related differential equations.[2]

References[]

  1. ^ Rédei, L. (1973), Lacunary polynomials over finite fields, translated by Földes, I., Elsevier, MR 0352060
  2. ^ a b Khovanskiĭ, A. G. (1991), Fewnomials, Translations of Mathematical Monographs, vol. 88, translated by Zdravkovska, Smilka, Providence, Rhode Island: American Mathematical Society, doi:10.1090/mmono/088, ISBN 0-8218-4547-0, MR 1108621
  3. ^ Roche, Daniel S. (2018), "What can (and can't) we do with sparse polynomials?", in Kauers, Manuel; Ovchinnikov, Alexey; Schost, Éric (eds.), Proceedings of the 2018 ACM on International Symposium on Symbolic and Algebraic Computation, ISSAC 2018, New York, NY, USA, July 16-19, 2018, Association for Computing Machinery, pp. 25–30, arXiv:1807.08289, doi:10.1145/3208976.3209027
  4. ^ Nakos, Vasileios (2020), "Nearly optimal sparse polynomial multiplication", IEEE Transactions on Information Theory, 66 (11): 7231–7236, arXiv:1901.09355, doi:10.1109/TIT.2020.2989385, MR 4173637
  5. ^ Pan, Victor Y. (2020), "Acceleration of subdivision root-finding for sparse polynomials", Computer algebra in scientific computing, Lecture Notes in Computer Science, vol. 12291, Cham: Springer, pp. 461–477, doi:10.1007/978-3-030-60026-6_27, MR 4184190
  6. ^ Zippel, Richard (1979), "Probabilistic algorithms for sparse polynomials", Symbolic and algebraic computation (EUROSAM '79, Internat. Sympos., Marseille, 1979), Lecture Notes in Computer Science, vol. 72, Berlin, New York: Springer, pp. 216–226, MR 0575692
  7. ^ Cohen, S. D.; Movahhedi, A.; Salinier, A. (1999), "Galois groups of trinomials", Journal of Algebra, 222 (2): 561–573, doi:10.1006/jabr.1999.8033, MR 1734229


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