So I'm looking for a function that takes in the degree of the polynomial and the range of coefficients from -c to c, and outputs a list of all the monic polynomials of that degree and with coefficients in that range.
I already have code to numerically compute the roots and plot in the complex plane, I just need a way to compute this list. I haven't been able to find previously posted code to do this task on stackexchange.
toMonicpol[lis_] :=
x^(Length[lis]) + Dot[lis, Table[x^r, {r, 0, Length[lis] - 1}]]
pols[deg_, c_] := Map[toMonicpol[#] &, Tuples[Range[-c, c], deg]]
pols[2, 3]
{-3 - 3 x + x^2, -3 - 2 x + x^2, -3 - x + x^2, -3 + x^2, -3 +
x + x^2, -3 + 2 x + x^2, -3 + 3 x + x^2, -2 - 3 x + x^2, -2 - 2 x +
x^2, -2 - x + x^2, -2 + x^2, -2 + x + x^2, -2 + 2 x + x^2, -2 +
3 x + x^2, -1 - 3 x + x^2, -1 - 2 x + x^2, -1 - x + x^2, -1 +
x^2, -1 + x + x^2, -1 + 2 x + x^2, -1 + 3 x + x^2, -3 x +
x^2, -2 x + x^2, -x + x^2, x^2, x + x^2, 2 x + x^2, 3 x + x^2,
1 - 3 x + x^2, 1 - 2 x + x^2, 1 - x + x^2, 1 + x^2, 1 + x + x^2,
1 + 2 x + x^2, 1 + 3 x + x^2, 2 - 3 x + x^2, 2 - 2 x + x^2,
2 - x + x^2, 2 + x^2, 2 + x + x^2, 2 + 2 x + x^2, 2 + 3 x + x^2,
3 - 3 x + x^2, 3 - 2 x + x^2, 3 - x + x^2, 3 + x^2, 3 + x + x^2,
3 + 2 x + x^2, 3 + 3 x + x^2}
Note that we can use FromDigits[{1, a, b, c}, x] to get a polynomial of degree Length[{a,b,c}] in x with leading coefficient 1:
Expand @ FromDigits[{1, a, b, c}, x]
c + b x + a x^2 + x^3
Using Tuples[Range[-c,c], deg], as in ulvi's answer, to generate all tuples (i.e., all possible non-leading coefficients in a polynomial with degree deg), and Prepending each tuple with 1, we can use FromDigits on each list to get the desired list of monic polynomials:
mp[d_, c_] := Expand @ FromDigits[Prepend[#, 1], x] & /@ Tuples[Range[-c, c], d];
This gives the same list of polynomials as ulvi's pols:
Sort[mp[2, 3] ] == Sort[pols[2, 3]]
True
