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# polynorm(?Expression, -NormPoly, -Residue)

Extracts and normalises the polynomial part of an arithmetic expression
*Expression*
- Arithmetic expression with constants and variables
*NormPoly*
- Normalized polynomial
*Residue*
- Residual nonpolynomial components in the form AuxVar=Expr

## Description

This predicate converts a general arithmetic expression into a
normalized polynomial representation and possibly a nonpolynomial
residue. The normalized polynomial representation is a follows:
A *monomial* is a list of constants and variables and represents
the product of all the list elements.
In a *normalised monomial*, the list is sorted, the first element is
the (only) constant and the others are variables.

A *normalised polynomial* is represented as a list of lists of
normalised monomials. The sublists represent groups of monomials
of the same degree in ascending order. If there are no monomials
for a certain degree, the list element is missing:

[ConstantMonos, LinearMonos, QuadraticMonos, CubicMonos, ...]

In a normalised polynomial, all monomials are normalised and
all monomials with identical variables are merged.
Non-polynomial components are factored out by introducing an
auxiliary variable and adding a term Aux=NonPolyExpr to the
Residue result list. All variables within Expression (which
are free at normalization time) are taken to be numerical
variables. If you intend to have variables which can be bound
to symbolic expressions rather than number, they must be
wrapped into an eval/1 functor.

## Examples

?- polynorm(2*5 + 3*(X+5*Y+7)*Z, Poly, Res).
X = X
Y = Y
Z = Z
Poly = [[[10]], [[21, Z]], [[3, X, Z], [15, Y, Z]]]
Res = []
yes.
?- polynorm(3*(X+Y), Poly, Res).
X = X
Y = Y
Poly = [[[3, X], [3, Y]]]
Res = []
yes.
?- polynorm(3, Poly, Res).
Poly = [[[3]]]
Res = []
yes.

## See Also

polydenorm / 2, polyrenorm / 2, linearize / 3