RegularChains[FastArithmeticTools]
NormalFormDim0
normal form of a polynomial w.r.t. a 0-dim regular chain
Calling Sequence
Parameters
Description
Examples
NormalFormDim0(f, rc, R)
R
-
polynomial ring
rc
a regular chain of R
f
polynomial of R
Returns the normal form of f w.r.t. rc in the sense of Groebner bases
rc must be a normalized zero-dimensional regular chain and all variables in f must be algebraic w.r.t. rc. See the subpackage ChainTools for these notions.
Moreover R must have a prime characteristic p such that FFT-based polynomial arithmetic can be used for this actual computation. The higher the degrees of f and rc are, the larger must be e such that 2e divides p−1. If the degree of f or rc is too large, then an error is raised.
The algorithm relies on the fast division trick (based on power series inversion) and FFT-based multivariate multiplication. When both commands NormalFormDim0 and NormalForm apply, the former one will often outperform the latter one.
with⁡RegularChains:
with⁡ChainTools:
with⁡FastArithmeticTools:
p≔962592769:
vars≔x,y,z:
N≔nops⁡vars:
R≔PolynomialRing⁡vars,p:
c1≔z2+286748183⁢z+134705551:
c2≔y2+914706789⁢y⁢z+686506773⁢y+823875308⁢z+417988453:
c3≔x2+224047618⁢x⁢y⁢z+197329999⁢x⁢y+838274835⁢x⁢z+792563861⁢x+600038529⁢y⁢z+434770098⁢y+251400283⁢z+918968185:
lf≔c1,c2,c3:
tc≔Chain⁡lf,Empty⁡R,R:
lf≔x+c1⁢c2⁢c3,c1−c2⁢c1−c3,1+c1⁢c2−c3:
dg≔10:
q1≔randpoly⁡vars,dense,degree=dg+rand⁡modpmodp:
q2≔randpoly⁡vars,dense,degree=dg+rand⁡modpmodp:
q3≔randpoly⁡vars,dense,degree=dg+rand⁡modpmodp:
r≔randpoly⁡vars,dense,degree=dg+rand⁡modpmodp:
f≔q1⁢c1+q2⁢c2+q3⁢c3+r
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