Generate terms of an m-order power series in n variables - wolfram-mathematica

Question

Consider a situation where you have data in a list of the form
data = {{x1, x2, x3, ..., xn, y}, {...}, ..., {...}}
For example,
data = {{0, 2, 3, 2}, {0, 0, 1, 4}, {7, 6, 8, 3}}
I'd like to fit the data to a multivariate polynomial of order, say, 2.
So, the 3-variable function values are:
{2, 4, 3}
in respective points
{{0, 2, 3}, {0, 0, 1}, {7, 6, 8}}
I'd say something like
Fit[data, {1, x, y, z, x^2, y^2, z^2, x y , x z, y z}, {x, y, z}]
This is all very nice, but i may not have only 3-variate data, there may be an arbitrary number of variables, and I don't know how to programmatically generate all the linear, quadratic or even higher-order terms, to insert them as the second argument of Fit[].
For 4-variate date do second order, it would be something like:
{1, x1, x2, x3, x4, x1^2, x2^2, x3^2, x4^2, x1 x2, x1 x3, x1 x4, x2 x3, x2 x4, x3 x4}
Is there any way I can generate such a list for n variables, to m-th order?
Like terms (without coefficients) in a m-order power series expansion of an n-variable function.

Answer 1

Does this do what you want?
Union[Times ### Tuples[{1, x, y, z}, 2]]

Answer 2

While the solution of #ruebenko is perfectly correct, I'd like to mention that it will be quite slow for higher powers / larger number of variables, because of the complexity of Tuples and lots of duplicates for higher powers. Here is an algebraic method with a much better performance for those cases (both run-time and memory-wise):
List ## Expand[(1 + x + y + z + t)^2] /. a_Integer*b_ :> b
Here is a comparison for large number of variables:
In[257]:= (res1=Union[Times###Tuples[{1,x,y,z,t},9]])//Short//Timing
Out[257]= {19.345,{1,t,t^2,t^3,t^4,t^5,t^6,t^7,t^8,t^9,x,<<694>>,x^2 z^7,y z^7,
t y z^7,x y z^7,y^2 z^7,z^8,t z^8,x z^8,y z^8,z^9}}
In[259]:= (res2=List##Expand[(1+x+y+z+t)^9]/. a_Integer*b_:>b)//Short//Timing
Out[259]= {0.016,{1,t,t^2,t^3,t^4,t^5,t^6,t^7,t^8,t^9,x,<<694>>,x^2 z^7,y z^7,
t y z^7,x y z^7,y^2 z^7,z^8,t z^8,x z^8,y z^8,z^9}}
In[260]:= res1===res2
Out[260]= True
In this case, we observe a 1000x speedup, but generally the two methods just have different computational complexities. The above code is an application of a general and nice method, called Algebraic Programming. For an interesting discussion of it in the context of Mathematica, see this Mathematica Journal paper by Andrzej Kozlowski.

Answer 3

Using #ruebenko's neat solution,
varsList[y_, n_?IntegerQ, k_?IntegerQ] :=
Union[Times ###
Tuples[Prepend[Table[Subscript[y, i], {i, 1, n}], 1], k]]
you can generate desired list via varsList[x, 4, 2].

Answer 4

Here is another method that I believe is worth knowing:
set = {1, x, y, z};
Union ## Outer[Times, set, set]