Irreducibility of Binomials

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.

Keywords:

Multivariate polynomial, irreducibility, unimodular matrix,

PDF

___

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.