Complete reduction modulo a polynomial ideal
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groebner::normalf(p, polys) computes a normal
form of the polynomial
p by complete reduction
modulo all polynomials in the list
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
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
the ideal generated by
respect to lexicographical ordering:
p := poly(x^2*y - 2*x*y + 1, [x,y]): groebner::normalf(p, [p1, p2], LexOrder);
p2 do not
form a Gröbner basis. The corresponding Gröbner basis leads
to a different normal form of
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 p1, …, pn, if and none of the leading terms of the pi divides the leading term of p, or if — for some i — g is a reduced form of p - q pi, where q is the quotient of the leading monomial of p and the leading monomial of pi. A reduced form always exists, but need not be unique. It is unique, if the pi form a Gröbner basis.
In the implementation of
reduction modulo some pi of
largest possible total degree is preferred, if reduction modulo several pi is