# (Real) algebraic geometry for (real) trigonometric polynomials?

Has somebody developed a comprehensive theory of the algebraic structure of trigonometric polynomials in several variables? If yes, where?

Background:

By a (real) trigonometric polynomial in $d$-variables, I mean a map $\mathbb{T}^d \to \mathbb{R}$ that is given by an expression of the form $$f(x) = \sum_{|k| \leq K} \hat{f}(k) \exp(2\pi\mathrm{i} k\cdot x)$$ where $k \in \mathbb{Z}^d$ and $|k| = \sup_{j=1,\dots,d} |k_j|$. Also $\mathbb{T} = \mathbb{R}/\mathbb{Z}$.

These trigonometric polynomials have many of the properties of usual polynomials, but are NOT polynomials. So as far as I know it, one cannot apply the usual algebraic-geometry constructions.

An example of a result, I would be interested in is: Given polynomials $f_1, \dots, f_{\ell}$ how does the dimension of their zero locus $$\{x \in \mathbb{T}^d:\quad f_j(x) = 0,\quad j=1,\dots,\ell\}$$ relate to the ideal generated by these polynomials?

One approach

In the Annals paper by Bourgain and Goldstein, a hint of how to do this is given. Write $$\exp(2\pi\mathrm{i} k \cdot x) = \prod_{j=1}^{d} \exp(2\pi\mathrm{i} x_j)^{k_j}.$$ Using that $\exp(2\pi\mathrm{i} x_j) = \cos(2\pi x_j) + \mathrm{i} \sin(2\pi x_j)$, one can write a trigonometric polynomial as a honest polynomial in the $2 d$ variables $C_j = \cos(2\pi x_j)$ and $S_j = \sin(2\pi x_j)$. A computation shows that this is a honest polynomial with real coefficients. Call this polynomial $\tilde{f}$.

These set from the previous example can then be described as the zero locus of the polynomials $\tilde{f}_j$ and the polynomials $$(C_j)^2 + (S_j)^2 = 1.$$

It seems to me that using this approach one can more or less carry over most results, but I am not very good at algebra, so I might miss subtleties. It would be nice if there was some work out of these things by somebody in the field.

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The answer is a definite yes, there is extensive literature addressing these. But it is somewhat scattered and I don't know that it represents a "field" per se, so it's hard to give precise pointers unless you have a specific question in mind. E.g. I've seen the $(C_j, S_j)$ technique you mention used in Engineering problems, Khovanskii has results about number of solutions... There's lots more that escapes me right now. – Thierry Zell Jun 3 '11 at 18:50
So one issue that puzzles me: Is there an analog of Bezout's theorem. So given 2 trigonometric polynomials $f$ and $g$ in two variables that don't have a common irreducible factor (as trigonometric polynomials). Is it true that the number of solutions of $f(x) = g(x) = 0$ is bounded by $4 \mathrm{deg}(f) \mathrm{deg}(g)$? ......... Also it would be interesting to see carefully stated versions of the claims about dimensions. – Helge Jun 3 '11 at 19:50

The Bézout theorem as you describe should work in $\mathbb{T}^2$ for the reasons you outlined (do the change of variables, add in the Pythagorean conditions and apply the regular Bézout). I am more familiar with the situation over $\mathbb{R}^2$, where, since the functions are periodic, you cannot expect finitely many solutions.