Function Repository Resource:

AlgebraicReplace

Source Notebook

Replace all occurrences of an algebraic subexpression with a new symbol

Contributed by: Daniel Lichtblau (Wolfram Research)

ResourceFunction["AlgebraicReplace"][expr,reps,repvars]

rewrites expr, replacing each occurrence of an element of reps with the corresponding element of repvars.

ResourceFunction["AlgebraicReplace"][expr,reps,repvars,vars]

rewrites expr, burrowing inside any subexpression that is not a polynomial in vars.

Details

As the name implies, ResourceFunction["AlgebraicReplace"] provides an algebraic form of polynomial replacement as opposed to requiring matching of patterns as used by ReplaceAll and related functions. For example, ordinary replacement of x y by z in the monomial x²y³ will have no effect, whereas AlgebraicReplace will return y z².

ResourceFunction["AlgebraicReplace"] uses PolynomialQ to determine whether to burrow into subexpressions. If vars is not specified, it will use Variables to determine them.

Examples

Basic Examples (1)

Algebraically replace xy by a new variable z in a bivariate polynomial:

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Scope (2)

AlgebraicReplace works with non-polynomial expressions:

In[2]:=

$ResourceFunction["AlgebraicReplace"][ Exp[3 x^3 - 7 x^2 y + 4 x y^2 + y^3] BesselJ[2, x^2 y^4]/ Sin[Pi x^4 y - 3], x y, z, {x, y}]$

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Use several replacements at once:

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$ResourceFunction["AlgebraicReplace"][ Exp[3 w^4 x^3 - 7 w x^2 y + 4 x y^2 + w^2 y^3] BesselJ[2, w^3 x^2 y + w x^2 y^4]/Sin[Pi w x^4 y - 3 E w x y z], {x w - y, x y}, {z, t}, {w, x, y}]$

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Properties and Relations (2)

ReplaceAll only replaces literal matches:

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AlgebraicReplace rewrites all monomials containing powers of xy to have powers of z:

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AlgebraicReplace with a set of reducing polynomials polys and replacement variables vars has behavior similar to PolynomialReduce with polys-vars:

In[6]:=

$poly = 3 w^4 x^3 - 7 w x^2 y + 4 x y^2 + w^2 y^3; reducers = {x y, w x^2 - y}; redvars = {z, t}; {ResourceFunction["AlgebraicReplace"][poly, reducers, redvars, {w, x, y}], PolynomialReduce[poly, reducers - redvars, {w, x, y, z, t}][[2]]}$

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This is not canonical insofar as the reduction is not done with a Gröbner basis, so results can depend on ordering of inputs:

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If we compute a Gröbner basis then the result is canonical for the given variable ordering:

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Reversing the reduction polynomials does not change the result of PolynomialReduce when a Gröbner basis is used:

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Possible Issues (2)

If the underlying variables are omitted, AlgebraicReplace might not recognize what are the correct ones:

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Specify x and y as variables to get the desired replacement:

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$ResourceFunction["AlgebraicReplace"][Sin[Pi x^4 y - 3], x y, z, {x, y}]$

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As seen in the "Properties and Relations" examples, AlgebraicReplace does not compute and use a Gröbner basis. Changing the order of the reducer inputs can affect the result.

Version History

1.0.1 – 26 May 2023
1.0.0 – 08 May 2023

Related Resources

Author Notes

This functionality is often requested on MSE, Wolfram Community and older forums as well. It could benefit from a better get-all-variables function. Another task for another day…

License Information

This work is licensed under a Creative Commons Attribution 4.0 International License