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Let $f : [0,2\pi]^d \to \mathbb R$ be multivariate trignometric polynomial of the form

$$ f(x_1, \cdots, x_d) = \sum_{i=1}^n a_i \prod_{j=1}^d f^{i}_j(x_j), \quad a_i \in \mathbb R $$

Where each $f^i_j$ is a trigonometric function, such as sine or cosine. I am interested in studying the (number of) connected components of the level sets of multivariate trigonometric polynomials.

Do you have any advice on relevant literature and available techniques? Is there something analogous to real algebraic geometry (smooth manifold theory in case the level sets are smooth manifolds) for multivariate trigonometric polynomials that could be used here? Any techniques available to upper bound the rank of $H^0$ (zeroth cohomology) of level sets defined by such multivariate polynomials.

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  • $\begingroup$ You could think of these as polynomials on $(S^1)^d$, i.e. plain old polynomials on $\mathbb R^{2d}$, then restricted so that each variable is unit length. From this perspective, what tools are you missing? $\endgroup$
    – Ryan Budney
    Commented Oct 13 at 0:46
  • $\begingroup$ I agree. This is a polynomial in $\mathbb R^{2d}$ at the end of the day. This is essentially one approach suggested here: mathoverflow.net/questions/66842/… I still want to compute bounds on the number of connected components. I am not sure if anything changes in this case. $\endgroup$
    – user82261
    Commented Oct 13 at 1:09
  • $\begingroup$ The exact form the polynomials take in my application might differ a bit but that's besides the point for now. $\endgroup$
    – user82261
    Commented Oct 13 at 1:09

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