This paper presents a new algorithm for computing the extended Hensel construction(EHC) of multivariate polynomials in main variable x and sub-variables u1, u2, ···, um over a number field K. This algor...This paper presents a new algorithm for computing the extended Hensel construction(EHC) of multivariate polynomials in main variable x and sub-variables u1, u2, ···, um over a number field K. This algorithm first constructs a set by using the resultant of two initial coprime factors w.r.t. x, and then obtains the Hensel factors by comparing the coefficients of xi on both sides of an equation. Since the Hensel factors are polynomials of the main variable with coefficients in fraction field K(u1, u2, ···, um), the computation cost of handling rational functions can be high. Therefore,the authors use a method which multiplies resultant and removes the denominators of the rational functions. Unlike previously-developed algorithms that use interpolation functions or Grobner basis, the algorithm relies little on polynomial division, and avoids multiplying by different factors when removing the denominators of Hensel factors. All algorithms are implemented using Magma, a computational algebra system and experiments indicate that our algorithm is more efficient.展开更多
Let D be a tame central division algebra over a Henselian valued field E,D be the residue division algebra of D,E be the residue field of E,and n be a positive integer.We prove that M_(n)(D)has a strictly maximal subf...Let D be a tame central division algebra over a Henselian valued field E,D be the residue division algebra of D,E be the residue field of E,and n be a positive integer.We prove that M_(n)(D)has a strictly maximal subfield which is Galois(resp.,abelian)over E if and only if M_(n)(D)has a strictly maximal subfield K which is Galois(resp.,abelian)and tame over E withГ_(K)■Г_(D),whereГ_(K)andГ_(D)are the value groups of K and D,respectively.This partially generalizes the result proved by Hanke et al.in 2016 for the case n=1.展开更多
基金supported in part by the National Natural Science Foundation of China under Grant No.11371356CAS Project QYZDJ-SSW-SYS022the Strategy Cooperation Project AQ-1701
文摘This paper presents a new algorithm for computing the extended Hensel construction(EHC) of multivariate polynomials in main variable x and sub-variables u1, u2, ···, um over a number field K. This algorithm first constructs a set by using the resultant of two initial coprime factors w.r.t. x, and then obtains the Hensel factors by comparing the coefficients of xi on both sides of an equation. Since the Hensel factors are polynomials of the main variable with coefficients in fraction field K(u1, u2, ···, um), the computation cost of handling rational functions can be high. Therefore,the authors use a method which multiplies resultant and removes the denominators of the rational functions. Unlike previously-developed algorithms that use interpolation functions or Grobner basis, the algorithm relies little on polynomial division, and avoids multiplying by different factors when removing the denominators of Hensel factors. All algorithms are implemented using Magma, a computational algebra system and experiments indicate that our algorithm is more efficient.
文摘Let D be a tame central division algebra over a Henselian valued field E,D be the residue division algebra of D,E be the residue field of E,and n be a positive integer.We prove that M_(n)(D)has a strictly maximal subfield which is Galois(resp.,abelian)over E if and only if M_(n)(D)has a strictly maximal subfield K which is Galois(resp.,abelian)and tame over E withГ_(K)■Г_(D),whereГ_(K)andГ_(D)are the value groups of K and D,respectively.This partially generalizes the result proved by Hanke et al.in 2016 for the case n=1.
