Complete reduction modulo a polynomial ideal

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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.


Example 1

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 p. In particular, if p is a polynomial expression, polys must be a list of polynomial expressions.


One of the identifiers DegInvLexOrder, DegreeOrder, and LexOrder, or a user-defined term ordering of type Dom::MonomOrdering. The default ordering is DegInvLexOrder.

Return Values

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 ig is a reduced form of p - qpi, 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 groebner::normalf, reduction modulo some pi of largest possible total degree is preferred, if reduction modulo several pi is possible.

See Also

MuPAD Functions