Sparse polynomial

In mathematics, a sparse polynomial (also lacunary polynomial^[1] or fewnomial)^[2] is a polynomial that has far fewer terms than its degree and number of variables would suggest. For example, ${\displaystyle x^{10}+3x^{3}+1}$ is a sparse polynomial as it is a trinomial with a degree of ${\displaystyle 10}$.

The motivation for studying sparse polynomials is to concentrate on the structure of a polynomial's monomials instead of its degree, as one can see, for instance, by comparing Bernstein-Kushnirenko theorem with Bezout's theorem. Research on sparse polynomials has also 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][5], division,^[6] root-finding algorithms,^[7] and polynomial greatest common divisors.^[8] 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.^[9]

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] Additionally, a sparse positivstellensatz exists for univariate sparse polynomials. It states that the non-negativity of a polynomial can be certified by sos polynomials whose degree only depends on the number of monomials of the polynomial.^[10]

Sparse polynomials oftentimes come up in sum or difference of powers equations. The sum of two cubes states that ${\displaystyle (x+y)(x^{2}-xy+y^{2})=x^{3}+y^{3}}$. Here ${\displaystyle x^{3}+y^{3}}$ is a sparse polynomial since out of the ${\displaystyle 16}$ possible terms, only ${\displaystyle 2}$ appear. Other examples include the identities ${\displaystyle (x-y)\sum _{k=0}^{N-1}x^{k}y^{N-1-k}=x^{N}-y^{N}}$ and also ${\displaystyle (x+y)\sum _{k=0}^{2n}(-1)^{k}x^{k}y^{2n-k}=x^{2n+1}+y^{2n+1}}$ where the product of two polynomials give a spearse polynomial. The Bring–Jerrard normal form of a quintic, ${\displaystyle x^{5}+px+q}$ is also a sparse polynomial.

See also

• Askold Khovanskii, one of the main contributors to the theory of fewnomials.

References

1. Rédei, L. (1973), Lacunary polynomials over finite fields, translated by Földes, I., Elsevier, MR 0352060
2. 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, S2CID 49868973
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, S2CID 59316578
5. Giorgi, Pascal; Grenet, Bruno; Perret du Cray, Armelle (2020), "Essentially optimal sparse polynomial multiplication", Proceedings of the 45th International Symposium on Symbolic and Algebraic Computation (ISSAC '20)., Association for Computing Machinery, pp. 202–209, arXiv:2001.11959, doi:10.1145/3373207.3404026, S2CID 211003922
6. Giorgi, Pascal; Grenet, Bruno; Perret du Cray, Armelle (2021), "On Exact Division and Divisibility Testing for Sparse Polynomials", Proceedings of the 2021 on International Symposium on Symbolic and Algebraic Computation (ISSAC '21)., Association for Computing Machinery, pp. 163–170, arXiv:2102.04826, doi:10.1145/3452143.3465539, S2CID 231855563
7. 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, S2CID 224820309
8. 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
9. 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
10. Averkov, Gennadiy; Scheiderer, Claus (2023-03-07). "Convex hulls of monomial curves, and a sparse positivstellensatz". arXiv:2303.03826 [math.OC].