Singular

5.1.33 factmodd

Syntax:

factmodd ( poly_expression, int_expression
[, poly_expression, poly_expression ]
[, int_expression, int_expression ]
)

Type:

list of polys

Purpose:

Computes a factorization of a polynomial h(x, y) in K[[x]][y] up to a certain degree in x, whenever a factorization of h(0, y) is provided or can be computed.
The algorithm is based on Hensel's lemma: Let h(x, y) denote a monic polynomial in y of degree m + n with coefficients in K[[x]]. Suppose there are two monic factors f_0(y) (of degree n) and g_0(y) of degree (m) such that
h(0, y) = f_0(y) * g_0(y) and <f_0, g_0> = K[y].
Fix an integer d >= 0. Then there are monic polynomials in y with coefficients in K[[x]], namely f(x, y) of degree n and g(x, y) of degree m such that
h(x, y) = f(x, y) * g(x, y) modulo <x^(d+1)> (*).
The function's six arguments are h, d, f_0, g_0, xIndex, and yIndex, where xIndex and yIndex denote indices of ring variables that are to play the roles of x and y as above. h must be provided as an element of K[x,y] since all terms of h with x-degree larger than d can be ignored due to (*).
If f_0 and g_0 are not given, the algorithm computes the factorization of h(0, y) and is expected to find exactly two distinct factors (which may appear with multiplicities larger than 1) and uses these as f_0 and g_0.
If xIndex and yIndex are missing they will be expected to be 1 and 2, respectively.

Note:

The function expects the ground ring to contain at least two variables.

Example:

ring r = 0, (x,y), dp; poly f0 = y240; poly g0 = y102+1; poly h = y342+14x260+7x140y110+2x120y130+y240; int d = 260; list L = factmodd(h, d, f0, g0); L; ==> [1]: ==> -14x260y204-4x240y224-14x260y102-7x140y212-2x120y232+14x260+7x140y110+\ 2x120y130+y240 ==> [2]: ==> 42x260y66+8x240y86+7x140y74+2x120y94+y102+1 // check result: next output should be zero reduce(h - L[1] * L[2], std(x^(d+1))); ==> 0