I need to find the diagonal polynomials induced to this equation: $$f(x_1,x_2,x_3,x_4)=x_1x_2+x_3x_4-x_1x_3+x_1x_2x_3x_4$$
Topic:These restrictions are counted as
1 variable: 1 polynomial.
2 variables: if one variable is x, and the other four are y, we have four polynomials; if two variables are x and the other three are y, we have $C_4^2 = $ polynomials.
3 variables: if one variable is x, another is y and the other three are z, the we have ?????????
4 variables: if one variable is x, another is y, another is z and the other two are w,then we have
For example, if we take $x_1 = x_2 = ... = x_4 = x $, with independent identical units, we get one diagonal (univariate) polynomial $$f(x)=x^2+x^4$$
What is the way to calculate the number of polynomials. Is there a better way or a theoretical basis?
Can anyone help me with clarification or references?
Thanks for the help.