About multivariable quadratic polynomials

Question

Say one has a polynomial function $f : \mathbb{C}^n \rightarrow \mathbb{C}$ such that it is quadratic in any of its variables $z_i$ (for $i \in \{ 1,2,..,n\}$). Then it follows that for any $i$ one can rewrite the polynomial as $f = A_i (z_i - a_i)(z_i - b_i)$, where the constants depend on the $i$ chosen.

• But does it also follow that one can write the function in the form, $f = \prod_{i=1}^n (B_i (z_i - c_i)(z_i - d_i) )$ ?

If the above is not true then what is the simplest decomposition that can be written for such a $f$?

• Like is it possible to redefine the basis in $\mathbb{C}^n$ to some $Z_i$ such that $f = \sum_{i} (a_i Z_i^2 + b_i Z_i + c_i)$ ?

• Like any extra leverage that can be gotten if one knows that $f$ is real-rooted?

Answer 1

You can't expect the $A_i,a_i,b_i$ to be constant. One could have $$f=z_1^2+z_2^2=(z_1+iz_2)(z_1-iz_2).$$

What about if $$f=z_1^2+z_2^2+z_3^2$$

Answer 2

In real algebraic geometry (a pretty hard subject), it is shown that a real-valued polynomial with real coefficients which is always non-negative can be written as sum of squares of ratios of polynomials. I'm not sure how "simple" you can get the representation if your original polynomial is both positive and negative valued.

The example $f(x,y) = x^2 + y^2$ shows that you always cannot hope for your multi-variable quadratic polynomial to be a product of univariate (one-variable) quadratics.

Also see the discussion in http://en.wikipedia.org/wiki/Hilbert%27s_seventeenth_problem to see references for explicit (but complicated) conditions under which a non-negative real valued polynomial can be written as a sum of squares of polynomials (instead of just ratios of polynomials).