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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