Enhancing the extended hensel construction by using Gröbner basis

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Contrary to that the general Hensel construction (GHC: [3]) uses univariate initial Hensel factors, the extended Hensel construction (EHC: [8]) uses multivariate initial Hensel factors determined by the Newton polygon (see below) of the given multivariate polynomial F (x, u ) ∈ K[ x , u ], where ( u ) = ( u 1 ,..., u ℓ ), with ℓ ≥ 2, and K is a number field. The F ( x, u ) may be such that its leading coefficient may vanish at ( u ) = ( 0 ) = (0,...,0), and even may be F ( x , 0 ) = 0. The EHC was used so far for computing series expansion of multivariate algebraic function determined by F ( x, u ) = 0, at critical points [8, 5] and for factorization [4, 1] and GCD computation [7] of F ( x , u ), without shifting the origin of u . It allows us to construct efficient algorithms for sparse multivariate polynomials [1, 7]. The EHC is another and promising approach than Zippel's sparse Hensel lifting [9, 10].