Proper formulation of Do expression

Question

I'm having trouble with the following code:

Do[
  P[x_, y_] := Sum[RandomChoice[{-1, 1}]*x^k*y^l, {k, 0, 3}, {l, 0, 3}],
  IrreduciblePolynomialQ[P[x, y]], 
  {12}]

Basically, I just want to generate multiple polynomials (in the above case 12) of the form

P[x_, y_] := Sum[RandomChoice[{-1, 1}]*x^k*y^l, {k, 0, 3}, {l, 0, 3}]

and have a list of whether these polynomials are irreducible. I don't actually need to see the equation of each individual polynomial, I just need a list of true, false so I can estimate some probabilities of irreducibility. Any help with this endeavor is immensely appreciated.

No need to put a definition, IrreduciblePolynomialQ[Sum[(* stuff *)]] works fine. You might want to use Table[] instead of Do[], tho. — J. M.'s missing motivation, Jun 15 '16 at 18:43

score 2 · Answer 1 · answered Jun 16 '16 at 02:40

Although I've told you that you can use a Sum[] expression within IrreduciblePolynomialQ[], it will be more expedient for you to use dot products; that is, generate your random coefficients as an array, and then take the dot product of that with a vector containing your basis functions.

Thus,

Table[IrreduciblePolynomialQ[
      (x^Range[0, 3]).RandomChoice[{-1, 1}, {4, 4}].(y^Range[0, 3])], {12}]

should generate the list you want, after which you can use either of Tally[] or Counts[] to help you compute your desired probabilities.

Proper formulation of Do expression

1 Answers1