Let
$$ f(x) = \sum\_{j=1}^n c\_j e^{2\pi i\alpha\_j x}, g(x) = \sum\_{k=1}^m d\_k e^{2\pi i\beta\_k x}$$
be two (quasi-periodic) trigonometric polynomials, where the coefficients $c\_j, d\_k$ are complex and the frequencies $\alpha\_j,\beta\_k$ are real (but are not necessarily commensurable). Suppose that $f,g$ have infinitely many zeroes in common, thus $f(x)=g(x)=0$ for infinitely many reals $x$. Does this imply that $f,g$ have a non-trivial common factor (in the ring of trigonometric polynomials)? In the commensurable case (so that $f,g$ have a common period) this is clear from the factor theorem, but my interest is in the incommensurable case. I would also be interested in the special case when $f,g$ are both real-valued, though I suspect that this restriction does not materially impact the question.


Here is a closely related question. Let $G = \{ (e^{2\pi i \alpha\_1 x}, \dots, e^{2\pi i \alpha\_k x}): x \in {\bf R} \}$ be a one-parameter subgroup of the torus $(S^1)^k$ for some reals $\alpha\_1,\dots,\alpha\_k$ that are linearly independent over ${\bf Q}$ (so that $G$ is dense in $(S^1)^k$). Is it true that every codimension two real subvariety of $(S^1)^k$ intersects $G$ in at most finitely many points? Note that if one specialises to the subvarieties $\{ z \in (S^1)^k: P(z)=Q(z)=0\}$ for two polynomials $P,Q$ with no common factor one basically obtains a version of the first question.


I'm somewhat aware of the literature on exponential polynomials (e.g., Ritt's theorem), but I couldn't find quite the appropriate tool; results in transcendental number theory (e.g., the [six exponentials theorem](https://en.wikipedia.org/wiki/Six_exponentials_theorem)) also seem vaguely relevant, but again it's not quite a perfect fit.



Is there a nullstellensatz for trigonometric polynomials?
I'm converting the [comment](https://mathoverflow.net/questions/387021/is-there-a-nullstellensatz-for-trigonometric-polynomials#comment986917_387021) above to an answer:


Shapiro made the following conjecture in the paper "[The expansion of mean-periodic functions in series of exponentials](https://doi.org/10.1002/cpa.3160110102)" Comm. Pure and Appl. Math. 11 (1958) 1–22:



> 
> If two exponential polynomials have infinitely many zeros in common they are both multiples of some third (entirely transcendental) exponential polynomial.
> 
> 
> 


In the paper "[On common zeros of exponential polynomials](https://www.e-periodica.ch/digbib/view?pid=ens-001%3A1975%3A21#169)" Enseignement Math. (2) 21 (1975), no. 1, 57–67, van der Poorten and Tijdeman prove a special case of Shapiro's conjecture for the case when one of the polynomials has rational frequencies. They observe that in a certain sense this statement is equivalent to the Skolem–Mahler–Lech theorem.


More recently, in the paper "[From Schanuel’s Conjecture to Shapiro’s Conjecture](http://doi.org/10.4171/CMH/328)", D'Aquino, Macintyre, and Terzo show that the general conjecture can be proven assuming Schanuel's conjecture.


