Complete reduction modulo a polynomial ideal
This functionality does not run in MATLAB.
groebner::normalf(p
, polys
, <order
>)
groebner::normalf(p, polys)
computes a normal
form of the polynomial p
by complete reduction
modulo all polynomials in the list polys
.
The rules laid down in the introduction to the groebner package concerning the polynomial types and the ordering apply.
The polynomials in the list polys
must all
be of the same type as p
. In particular, do not
mix polynomials created via poly
and
polynomial expressions.
We consider the ideal generated by the following polynomials:
p1 := poly(x^2  x + 2*y^2, [x,y]): p2 := poly(x + 2*y  1, [x,y]):
We compute the normal form of the following polynomial p
modulo
the ideal generated by p1
, p2
with
respect to lexicographical ordering:
p := poly(x^2*y  2*x*y + 1, [x,y]): groebner::normalf(p, [p1, p2], LexOrder);
Note that p1
, p2
do not
form a Gröbner basis. The corresponding Gröbner basis leads
to a different normal form of p
:
groebner::normalf(p, groebner::gbasis([p1, p2]), LexOrder)
delete p1, p2, p:

A polynomial or a polynomial expression. The coefficients in this polynomial and polynomial expression can be arbitrary arithmetical expressions. 

A list of polynomials of the same type as 

One of the identifiers 
Polynomial of the same type as the input polynomials. If polynomial expressions are used as input, then a polynomial expression is returned.
A polynomial g is a reduced form of a polynomial p modulo a list of polynomials p_{1}, …, p_{n}, if and none of the leading terms of the p_{i} divides the leading term of p, or if — for some i — g is a reduced form of p  q p_{i}, where q is the quotient of the leading monomial of p and the leading monomial of p_{i}. A reduced form always exists, but need not be unique. It is unique, if the p_{i} form a Gröbner basis.
In the implementation of groebner::normalf
,
reduction modulo some p_{i} of
largest possible total degree is preferred, if reduction modulo several p_{i} is
possible.