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Year 2023, Volume: 34 Issue: 34, 62 - 70, 10.07.2023
https://doi.org/10.24330/ieja.1260484

Abstract

References

  • 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

Year 2023, Volume: 34 Issue: 34, 62 - 70, 10.07.2023
https://doi.org/10.24330/ieja.1260484

Abstract

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.

References

  • 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.
There are 17 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Haohao Wang This is me

Jerzy Wojdylo This is me

Peter Oman This is me

Early Pub Date May 11, 2023
Publication Date July 10, 2023
Published in Issue Year 2023 Volume: 34 Issue: 34

Cite

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. July 2023;34(34):62-70. doi:10.24330/ieja.1260484
Chicago Wang, Haohao, Jerzy Wojdylo, and Peter Oman. “Irreducibility of Binomials”. International Electronic Journal of Algebra 34, no. 34 (July 2023): 62-70. https://doi.org/10.24330/ieja.1260484.
EndNote Wang H, Wojdylo J, Oman P (July 1, 2023) Irreducibility of Binomials. International Electronic Journal of Algebra 34 34 62–70.
IEEE H. Wang, J. Wojdylo, and P. Oman, “Irreducibility of Binomials”, IEJA, vol. 34, no. 34, pp. 62–70, 2023, doi: 10.24330/ieja.1260484.
ISNAD Wang, Haohao et al. “Irreducibility of Binomials”. International Electronic Journal of Algebra 34/34 (July 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 et al. “Irreducibility of Binomials”. International Electronic Journal of Algebra, vol. 34, no. 34, 2023, pp. 62-70, doi:10.24330/ieja.1260484.
Vancouver Wang H, Wojdylo J, Oman P. Irreducibility of Binomials. IEJA. 2023;34(34):62-70.