# List all degree one, two, three polynomials.

All degree one polynomials over $\mathbb{C}$ are of the form $ax+b$, where $a,b$ are complex numbers.

How two write down all degree two (and three) polynomials over $\mathbb{C}$? I only need to write down normal forms of the polynomials. Any help will be greatly appreciated!

Edit: sorry, I forgot to say that the polynomials have three variables. My question is the following.

All degree one polynomials with three variables over $\mathbb{C}$ are of the form $a_1x_1 + a_2 x_2 + a_3 x_3 +b$, where $a_1, a_2, a_3, b$ are complex numbers.

How two write down all degree two (and three) polynomials over $\mathbb{C}$? I only need to write down normal forms of the polynomials. Any help will be greatly appreciated!

You can write them down explicitly, such as for degree 2: $$a x_1^2 + b x_2^2 + c x_3^2 + d x_1 x_2 + e x_1 x_3 + f x_2 x_3 + g x_1 + h x_2 + i x_3 + j$$ But that gets tedious. Therefore I advise using sum and product notation. For degree $d$ and three variables, you have: $$\sum_{i=0}^{d} \sum_{j=0}^{d-i} \sum_{k=0}^{d-i-j} a_{ijk} x_1^i x_2^j x_3^k$$ Note that this covers all degrees in a single formula. For example with $d=1$ you simply get $$a_{100} x_1 + a_{010} x_2 + a_{001} x_2 + a_{000}$$ and for degree 2, it gives $$a_{200} x_1^2 + a_{020} x_2^2 + a_{002} x_3^2 + a_{110} x_1 x_2 + a_{101} x_1 x_3 + a_{011} x_2 x_3 + a_{100} x_1 + a_{010} x_2 + a_{001} x_3 + a_{000}$$
To get the degree one polynomials, you just looked at $\mathbb{C}$-linear combinations of all the degree one monomials, $x_1$, $x_2$ and $x_3$, plus the constant (degree zero) term. The degree two and three polynomials work the same. To find all degree two polynomials just look at all $\mathbb{C}$-linear combinations of all the degrees zero/one/two monomials. The degree two monomials are $x_1^2$, $x_2^2$, $x_3^2$, $x_1x_2$, $x_1x_3$, $x_2x_3$. So the degree two polynomials are $$a+bx_1+cx_2+dx_3+ex_1^2+fx_2^2+gx_3^2+hx_1x_2+kx_1x_3+lx_2x_3$$ where the coefficients are in $\mathbb{C}$ and at least one of $e,f,g,h,k,l$ are not zero.