Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.
I think if you have convergence for all $x$ in $\mathbb R^n$, you will have bounds on the coefficients that would make it absolutely convergent for all $x$ in $\mathbb C^n$. An algebraic function will have at most polynomial growth, so Cauchy's theorem should imply that high degree coefficients vanish.
It is not surprising that the best polynomials differ just a bit at the end. If you change a few of the lower terms in a polynomial, then roots $\xi$ with $|\xi|$ noticeably larger than one will be only mildly affected.
Just a comment: once you have a large root of a polynomial, you can multiply by $(1\pm z^n)$ to another polynomial of twice the length with the same root.
If you look at weight two cusp forms for the principal congruence subgroup $\Gamma(7)$ of $SL_2(\mathbb Z)$, they form a dimension three representation. This seems geometric enough to me.
"Variations of the mixed Hodge structure of affine hypersurfaces in algebraic tori" by Victor Batyrev is the original reference. The bottom line is that X inherits singularities from the ambient space, in the sense that it looks, local analytically, as a disc times a toric singularity.
It sure seems like a rather silly question. However, a priori one could say the same thing about the sequence of factorials. So I will upvote it, just for kicks...
