# Attributes for private functions

I am not sure if what I want to do can be done, but I thought to ask anyway.

Note: I already have in hand a workaround, I was just wondering for the power/simplicity that Mathematica may provide.

Problem: Suppose that I have in hand a polynomial in x, y, z and only products between x,, z and y, z occur, e.g.

p = Pi x*z + Zeta x^2*z + 123/7 z + 1/2 x + 5 y + 9/2 y*z + 3 y^5*z + 1/Pi* z^4,

where I put some random constant coefficients in front of every term.

Now suppose, I have a homemade function that only executes "stuff" on numerical/numbers such as Zeta or 19/2. If I apply that function (e.g. myF) to p (myF /@ p) then it doesn't do "the job" because it can't handle the variables x,y,z.

Is there a way when I do myF /@ p to actually get

myF /@ p = myF[Pi] x*z + myF[Zeta] x^2*z + myF[123/7] z + myF[1/2] x + myF y + myF[9/2] y*z + myF y^5*z + myF[1/Pi]* z^4


I know that the build in function N, has attribute NHoldAll and this is how it performs a similar to the problem, job.

Comment: No I will not give you more information (!) about the function myF because I want this to be as generic as it gets. If it has a solution, it is nice to be known and applied to any kind of "home-brewed" function since I can think of a big class of functions that would a enjoy such an attribute.

Thank you!

• Have you already seen Distribute[]? – J. M.'s torpor May 6 '20 at 9:20
• Would you care to elaborate, possibly with a small snippet? Thank you. – hal May 6 '20 at 9:39
• You did not operate on the number 2 in x^2 in the second term. I suppose you mean to operate only on the numerical coefficients? Might nonnumerical coefficients arise? Are the polynomials always expanded as in the example? I suppose the coefficient zero should be mapped to zero, not myF? – Michael E2 May 6 '20 at 12:10
• Further, what about the coeffiicient 1? – Michael E2 May 6 '20 at 12:31
• "I just want the function to be able to 'pick up' the constants in front of the polynomials..." -- And I was taught in school that the constant in front of x^2 was to be treated as a 1. Which way do you want it, x^2 becomes myF x^2 or it remains x^2? – Michael E2 May 6 '20 at 19:23

## 1 Answer

Perhaps either

Fold[#1.{x, y, z} + #2 &,
Function[c, If[c == 0 || c == 1, c, myF[c]], Listable]@
Reverse@CoefficientArrays[p, {x, y, z}]]


or

FromCoefficientRules[
Normal[myF /@ Association@CoefficientRules[p, {x, y, z}]],
{x, y, z}]


[Update: Changed first code to leave 1 (as well as 0) alone; unsure if the comment is saying that's what's desired. Otherwise, I would appreciate an example where the above fails.]

• Ok I just tried it and it works. It is a neat solution, thank you :). If you are kind enough, perhaps you could provide me with some clarifications: Why the Reverse? What's its purpose? – hal May 7 '20 at 7:51
• @hal CoefficientArrays[] returns the coefficients as a list of arrays according to the degree, starting with the constant coefficient and ending with highest-degree terms. The polynomial is constructed from the coefficients using a multivariate form of Horner's rule starting from the highest-degree coefficients. (So Reverse puts the highest-degree coefficient first.) – Michael E2 May 7 '20 at 12:30