For a multivariable polynomial of degree t, how can you find the coefficients of monomials of degree t?

Question

Hello Mathematica StackExchange community,

I'm hoping that someone can be of help!

Let p be a fixed prime. I have a multivariable polynomial g (of degree t). I need a list of all the monomial terms of g which satisfy:

1. Degree of the monomial term is t.
2. Each of the exponents in the monomial term is less than or equal to p-2

I have been playing around with the Mathematica commands (MonomialList, Coefficient, CoefficientRules, Expand, etc.), but have not been successful in reaching my goal.

If anybody can provide some help, it is greatly appreciated! Thank you in advance.

Sincerely, Richard M. Low

UPDATE:---------------------------------

Due to kglr's help, I have been able to make some progress on my problem. Here is kglr's original Mathematica code:

ClearAll[x, y, z, f]
t = 4;
p = 5;
poly = (x + 2 y + z)^t + x^2;

f = Select[And[Total@# == t, Max[#] <= p - 2] &@Exponent[#, {x, y, z}] &];

selected = f@MonomialList[poly]

I'm hoping that kglr (or somebody else) can help me modify it slightly.

Since the typical polynomials that I will be working with may have (literally) billions (~ 6^14) of monomial terms, I would like for the output of f to be 1 monomial term (not the entire list) which satisfies the two conditions (in my original post) AND whose coefficient is not equal to 0 (mod p).

This will greatly decrease the runtime and memory usage of the Mathematica program. The runtime of the original Mathematica program went past several days and I ended up aborting the calculation.

Again, I appreciate any help that the Mathematica StackExchange can provide. I am completely incompetent in Mathematica programming.

Thank you in advance.

Sincerely, Richard M. Low

Answer 1

ClearAll[x, y, z, f]
t = 4;
p = 5;
poly = (x + 2 y + z)^t + x^2;

f = Select[And[Total@# == t, Max[#] <= p - 2] &@Exponent[#, {x, y, z}] &];

selected = f @ MonomialList[poly]

{8 x^3 y, 4 x^3 z, 24 x^2 y^2, 24 x^2 y z, 6 x^2 z^2, 32 x y^3, 48 x y^2 z, 24 x y z^2, 4 x z^3, 32 y^3 z, 24 y^2 z^2, 8 y z^3}

Complement[MonomialList @ poly, selected]

{x^2, x^4, 16 y^4, z^4}

Update: "I would like for the output of f to be 1 monomial term (not the entire list) which satisfies the two conditions."

Just replace Select with SelectFirst:

f2 = SelectFirst[And[And[Total@# == t, Max[#] <= p - 2] &@Exponent[#, {x, y, z}], 
  Mod[# /. x | y | z -> 1, p] != 0] &]

selected2 = f2@MonomialList[poly]

8 x^3 y

Update 2: A faster approach: use Expand instead of MonomialList:

f2 @ Expand[poly]

8 x^3 y

Answer 2

It is also possible to use the output of CoefficientRules[] to get what is wanted. To wit,

With[{t = 4, p = 5},
     KeyValueMap[#2 Inner[Power, {x, y, z}, #1, Times] &, 
                 KeySelect[CoefficientRules[(x + 2 y + z)^t + x^2, {x, y, z}], 
                           Tr[#] == t && Max[#] <= p - 2 &]]]
   {8 x^3 y, 4 x^3 z, 24 x^2 y^2, 24 x^2 y z, 6 x^2 z^2, 32 x y^3, 48 x y^2 z,
    24 x y z^2, 4 x z^3, 32 y^3 z, 24 y^2 z^2, 8 y z^3}

To add in the further $\bmod p$ restriction, you can use another call to Select[]:

With[{t = 4, p = 5},
     KeyValueMap[#2 Inner[Power, {x, y, z}, #1, Times] &, 
                 Drop[Select[KeySelect[CoefficientRules[(x + 2 y + z)^t + x^2,
                                                        {x, y, z}], 
                                       Tr[#] == t && Max[#] <= p - 2 &],
                             ! Divisible[#, p] &], {2, -1}]]]
   {8 x^3 y}

where I use the explicit test Divisible[] for filtering.