Araştırma Makalesi
BibTex RIS Kaynak Göster
Yıl 2023, Cilt: 34 Sayı: 34, 62 - 70, 10.07.2023
https://doi.org/10.24330/ieja.1260484

Öz

Kaynakça

  • C. de Boor, Polynomial interpolation in several variables, in Studies in Computer Science (in Honor of Samuel D. Conte), R. DeMillo and J. R. Rice (eds.), (1994), Plenum Press New York, 87-119.
  • J. Bruening and H. Wang, An implicit equation given certain parametric equations, Missouri J. Math. Sci., 18(3) (2006), 213-220.
  • S. Gao, Absolute irreducibility of polynomials via Newton polytopes, J. Algebra, 237(2) (2001), 501-520.
  • S. Gao, Factoring multivariate polynomials via partial differential equations, Math. Comp., 72(242) (2003), 801-822.
  • S. Gao and A. G. B. Lauder, Decomposition of polytopes and polynomials, Discrete Comput. Geom., 26(1) (2001), 89-104.
  • S. Gao and A. G. B. Lauder, Hensel lifting and bivariate polynomial factorisation over finite fields, Math. Comput., 71(240) (2002), 1663-1676.
  • D. Hilbert,  Uber die Theorie der algebraischen Formen, Math. Ann., 36 (1890), 473-534.
  • J. W. Hoffman and H. Wang, A study of a family of monomial ideals, J. Algebra Appl., 22(3) (2023), 2350068 (23 pp).
  • T. Hungerford, Algebra, Springer-Verlag, New York, 1974.
  • D. Inaba, Factorization of multivariate polynomials by extended hensel construction, SIGSAM Bull., 39(1) (2005), 2-14.
  • S. M. M. Javadi and M. B. Monagan, On factorization of multivariate polynomialsover algebraic number and function fields, In Proceedings of the 2009international symposium on Symbolic and algebraic computation, ISSAC 2009,New York, NY, USA, (2009), 199-206.
  • E. Kaltofen, J. P. May, Z. Yang and L. Zhi, Approximate factorization of multivariate polynomials using singular value decomposition, J. Symbolic Comput., 43(5) (2008), 359-376.
  • K. S. Kedlaya and C. Umans, Fast polynomial factorization and modular composition, SIAM J. Comput., 40(6) (2011), 1767-1802.
  • Z. Mou-Yan and R. Unbehauen, Approximate factorization of multivariable polynomials, Signal Process, 14(2) (1988), 141-152.
  • T. Sasaki, Approximate multivariate polynomial factorization based on zerosum relations, In Proc. ISSAC2001, ACM Press, (2001), 284-291.
  • J. Von Zur Gathen, Irreducibility of multivariate polynomials, J. Comput. System Sci., 31(2) (1985), 225-264.
  • W. Wu and Z. Zeng, The numerical factorization of polynomials, Found. Comput. Math., 17(1) (2015), 259-286.

Irreducibility of Binomials

Yıl 2023, Cilt: 34 Sayı: 34, 62 - 70, 10.07.2023
https://doi.org/10.24330/ieja.1260484

Öz

In this paper, we prove that the family of binomials $x_1^{a_1}
\cdots x_m^{a_m}-y_1^{b_1}\cdots y_n^{b_n}$ with $\gcd(a_1,
\ldots, a_m, b_1, \ldots, b_n)=1$ is irreducible by identifying
the connection between the irreducibility of a binomial in
${\mathbb C}[x_1, \ldots, x_m, y_1, \ldots, y_n]$ and ${\mathbb
C}(x_2, \ldots, x_m, y_1, \ldots, y_n)[x_1]$. Then we show that
the necessary and sufficient conditions for the irreducibility of
this family of binomials is equivalent to the existence of a
unimodular matrix $U_i$ with integer entries such that $(a_1,
\ldots, a_m, b_1, \ldots, b_n)^T=U_i \be_i$ for $i\in \{1, \ldots,
m+n\}$, where $\be_i$ is the standard basis vector.

Kaynakça

  • C. de Boor, Polynomial interpolation in several variables, in Studies in Computer Science (in Honor of Samuel D. Conte), R. DeMillo and J. R. Rice (eds.), (1994), Plenum Press New York, 87-119.
  • J. Bruening and H. Wang, An implicit equation given certain parametric equations, Missouri J. Math. Sci., 18(3) (2006), 213-220.
  • S. Gao, Absolute irreducibility of polynomials via Newton polytopes, J. Algebra, 237(2) (2001), 501-520.
  • S. Gao, Factoring multivariate polynomials via partial differential equations, Math. Comp., 72(242) (2003), 801-822.
  • S. Gao and A. G. B. Lauder, Decomposition of polytopes and polynomials, Discrete Comput. Geom., 26(1) (2001), 89-104.
  • S. Gao and A. G. B. Lauder, Hensel lifting and bivariate polynomial factorisation over finite fields, Math. Comput., 71(240) (2002), 1663-1676.
  • D. Hilbert,  Uber die Theorie der algebraischen Formen, Math. Ann., 36 (1890), 473-534.
  • J. W. Hoffman and H. Wang, A study of a family of monomial ideals, J. Algebra Appl., 22(3) (2023), 2350068 (23 pp).
  • T. Hungerford, Algebra, Springer-Verlag, New York, 1974.
  • D. Inaba, Factorization of multivariate polynomials by extended hensel construction, SIGSAM Bull., 39(1) (2005), 2-14.
  • S. M. M. Javadi and M. B. Monagan, On factorization of multivariate polynomialsover algebraic number and function fields, In Proceedings of the 2009international symposium on Symbolic and algebraic computation, ISSAC 2009,New York, NY, USA, (2009), 199-206.
  • E. Kaltofen, J. P. May, Z. Yang and L. Zhi, Approximate factorization of multivariate polynomials using singular value decomposition, J. Symbolic Comput., 43(5) (2008), 359-376.
  • K. S. Kedlaya and C. Umans, Fast polynomial factorization and modular composition, SIAM J. Comput., 40(6) (2011), 1767-1802.
  • Z. Mou-Yan and R. Unbehauen, Approximate factorization of multivariable polynomials, Signal Process, 14(2) (1988), 141-152.
  • T. Sasaki, Approximate multivariate polynomial factorization based on zerosum relations, In Proc. ISSAC2001, ACM Press, (2001), 284-291.
  • J. Von Zur Gathen, Irreducibility of multivariate polynomials, J. Comput. System Sci., 31(2) (1985), 225-264.
  • W. Wu and Z. Zeng, The numerical factorization of polynomials, Found. Comput. Math., 17(1) (2015), 259-286.
Toplam 17 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Matematik
Bölüm Makaleler
Yazarlar

Haohao Wang Bu kişi benim

Jerzy Wojdylo Bu kişi benim

Peter Oman Bu kişi benim

Erken Görünüm Tarihi 11 Mayıs 2023
Yayımlanma Tarihi 10 Temmuz 2023
Yayımlandığı Sayı Yıl 2023 Cilt: 34 Sayı: 34

Kaynak Göster

APA Wang, H., Wojdylo, J., & Oman, P. (2023). Irreducibility of Binomials. International Electronic Journal of Algebra, 34(34), 62-70. https://doi.org/10.24330/ieja.1260484
AMA Wang H, Wojdylo J, Oman P. Irreducibility of Binomials. IEJA. Temmuz 2023;34(34):62-70. doi:10.24330/ieja.1260484
Chicago Wang, Haohao, Jerzy Wojdylo, ve Peter Oman. “Irreducibility of Binomials”. International Electronic Journal of Algebra 34, sy. 34 (Temmuz 2023): 62-70. https://doi.org/10.24330/ieja.1260484.
EndNote Wang H, Wojdylo J, Oman P (01 Temmuz 2023) Irreducibility of Binomials. International Electronic Journal of Algebra 34 34 62–70.
IEEE H. Wang, J. Wojdylo, ve P. Oman, “Irreducibility of Binomials”, IEJA, c. 34, sy. 34, ss. 62–70, 2023, doi: 10.24330/ieja.1260484.
ISNAD Wang, Haohao vd. “Irreducibility of Binomials”. International Electronic Journal of Algebra 34/34 (Temmuz 2023), 62-70. https://doi.org/10.24330/ieja.1260484.
JAMA Wang H, Wojdylo J, Oman P. Irreducibility of Binomials. IEJA. 2023;34:62–70.
MLA Wang, Haohao vd. “Irreducibility of Binomials”. International Electronic Journal of Algebra, c. 34, sy. 34, 2023, ss. 62-70, doi:10.24330/ieja.1260484.
Vancouver Wang H, Wojdylo J, Oman P. Irreducibility of Binomials. IEJA. 2023;34(34):62-70.