       Re: MonomialQ

• To: mathgroup at smc.vnet.net
• Subject: [mg33559] Re: [mg33542] MonomialQ
• From: Rob Pratt <rpratt at email.unc.edu>
• Date: Sun, 31 Mar 2002 04:09:08 -0500 (EST)
• Sender: owner-wri-mathgroup at wolfram.com

```On Fri, 29 Mar 2002, Detlef Mueller wrote:

> Hello,
>
> I want to test, wether an expression is a Monomial
> with respect to a List of Variables.
> Here a Monomial is defined as some coefficient
> multiplied by a product of powers of the
> Variables: c*v1^p1*... vn^pn,
> where c must not depend on any of v1,..vn.
> (this is sometimes called "term").
>
> Similar to the build in Function PolynomialQ.
>
> Say
>
> MonomialQ[(a b)^3/(b c),{a,b}] -> True
> MonomialQ[(a b)^3/(b c),{a,c}] -> False
> MonomialQ[(a b)^3/(b c)-c,{a,b}] -> False
> MonomialQ[(a+b)(a+c),{a}] -> False
> MonomialQ[(a+b)(a+c)-(a^2+bc),{a}]->True
>
> My current Implementation is
>
> MonomialQ[a_,L_List] :=
>     PolynomialQ[a, L] &&               (* must be Polynomial and *)
>       Head[Expand[a, x_ /; MemberQ[L, x]]] =!= Plus;
>
> Maybe, there is no better way, but I wonder, if
> the Expand with Pattern is a good Idea here.
>
> Since in most Cases "a" _is_ in fact a Monomial
> in simple power-product-form, and since this is
> an often used Function, using "Expand" might be
> nearly every time an Overkill.
>
> Maybe there is a quick "pretest", checking
> for the "simple power-product-form"
> a = a1^k1*...an^kn, a1,..an Symbols or Constants,
> k1..kn non negative integers?
>
> Greetings,
>   Detlef

After checking that a is a polynomial in the variables L, you can check
that setting everything in L to 0 yields 0.

MonomialQ[a_, L_List] :=
PolynomialQ[a, L] && (a /. (#1 -> 0 & ) /@ L) === 0

It does give the desired results for your examples (after changing bc to

Rob Pratt
Department of Operations Research
The University of North Carolina at Chapel Hill

rpratt at email.unc.edu

http://www.unc.edu/~rpratt/

```

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