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, is a sparse polynomial, as it is a trinomial with a degree of .

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 often come up in sum or difference of powers equations. The sum of two cubes states that . Here is a sparse polynomial, since out of the possible terms, only appear. Other examples include the identities and also where the product of two polynomials give a spearse polynomial. The Bring–Jerrard normal form of a quintic, is also a sparse polynomial.

See also

edit

References

edit
  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, ISBN 978-1-4503-5550-6, S2CID 49868973
  4. ^ Nakos, Vasileios (2020), "Nearly optimal sparse polynomial multiplication", IEEE Transactions on Information Theory, 66 (11): 7231–7236, arXiv:1901.09355, Bibcode:2020ITIT...66.7231N, 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, ISBN 978-1-4503-7100-1, 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, ISBN 978-1-4503-8382-0, 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, ISBN 978-3-030-60025-9, 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].


Stub icon

This polynomial-related article is a stub. You can help Wikipedia by adding missing information.

📚 Artikel Terkait di Wikipedia

Binomial (polynomial)

a binomial is a polynomial that is the sum of two terms, each of which is a monomial. It is the simplest kind of a sparse polynomial after the monomials

Computation of cyclic redundancy checks

the CRC of the message modulo a sparse polynomial which is a multiple of the CRC polynomial. For CRC-32, the polynomial x123 + x111 + x92 + x84 + x64 +

Polynomial identity testing

runtime. A sparse PIT has at most m {\displaystyle m} nonzero monomial terms. A sparse PIT can be deterministically solved in polynomial time of the

Xorshift

efficient implementation in software without the excessive use of sparse polynomials. They generate the next number in their sequence by repeatedly taking

Monomial

In mathematics, a monomial is, roughly speaking, a polynomial which has only one term. Two definitions of a monomial may be encountered: A monomial, also

Polynomial greatest common divisor

GCD or gcd) of two polynomials is a polynomial, of the highest possible degree, which is a factor of both the two original polynomials. This concept is

Sparse binary polynomial hashing

Sparse binary polynomial hashing (SBPH) is a generalization of Bayesian spam filtering that can match mutating phrases as well as single words. SBPH is

Lacuna

measure of the extent that a pattern contains gaps Lacunary polynomial, or sparse polynomial Petrovsky lacuna, in mathematics Laguna (disambiguation) This