Create a polynomial of a given degree

polynomial[vars_List, n_Integer, coeff_] :=
   #.Array[coeff, Length@#] &@ DeleteDuplicates[Times @@@ Tuples[Prepend[vars, 1], n]]

Clear[a]
polynomial[{x, y, z}, 3, a]
(* a[1] + x a[2] + y a[3] + z a[4] + x^2 a[5] + x y a[6] + x z a[7]
     + y^2 a[8] + y z a[9] + z^2 a[10] + x^3 a[11] + x^2 y a[12]
     + x^2 z a[13] + x y^2 a[14] + x y z a[15] + x z^2 a[16]
     + y^3 a[17] + y^2 z a[18] + y z^2 a[19] + z^3 a[20] *)

for a one liner

poly[{x_,y_,z_},n_,a_]:= Sum[a[i, j, k] x^i y^j z^k,
                              {i, 0, n}, {j, 0, n - i}, {k, 0, n - i - j}]

another way

poly[vars_List, a_, order_] := Module[{n = Length@vars, idx, z},
  idx = Cases[Tuples[Range[0, order], n], x_ /; Plus @@ x <= order];
  z = Times @@@ (vars^# & /@ idx);
  z.((Subscript[a, Row[#]]) & /@ idx)
  ]

poly[{x, y, z}, a, 3]  (*a is used for coefficient*)

poly[{x, y, z}, a, 2]

poly[{x, y}, a, 2]

poly[{x}, a, 4]

poly[{x, y, z, w}, a, 5]

To make M display the coeffs first, use ParameterVariables :> {a} with TraditionalForm (this is for display only)

TraditionalForm[poly[{x, y, z}, a, 3], ParameterVariables :> {a}]

ps. If you do not like to use subscripts, you can use this instead:

poly[vars_List, order_] := Module[{n = Length@vars, idx, z},
  idx = Cases[Tuples[Range[0, order], n], x_ /; Plus @@ x <= order];
  z = Times @@@ (vars^# & /@ idx);
  z.((ToExpression["a" <> ToString[Row[#]]]) & /@ idx)
  ]

and now poly[{x, y, z}, 3] gives