I am having difficulty finding the right sources/papers to look through for the complexity class of deciding if a Variety over C is non-empty.
I can show that saying a Variety is empty is in NP, since one can compute a Grobner basis and check if it's 1, then the verifier simply takes as input the summation showing that 1 is in the ideal.
But I can't find results for the complexity class of the complement problem. I can reduce the problem to finding a $\bar{Q}$-rational point in the algebraic variety, since to confirm it all we have to do is have a point which we can do polynomial-time computations with and confirm that it is in the variety. However, finding rational points seems difficult from my cursory look through the Wikipedia page. Along these lines, I have heard that numerical algebraic geometry has a method of sampling points (randomly?) in variety - maybe this would help me? But I don't understand that process.
In Solving polynomial equations; Foundations, algorithms, and applications the authors state on pg. 260
Algebraic variety seems to be the standard definition of a variety in affine space, and encoded in dense form is defined in a previous section and simply needs that our polynomials are of bounded degree. This seems really powerful, but the original paper is in French and I can't read French. Maybe someone more familiar with this subject could help me make sure I understand this result correctly?
Here is a link to the GH93 paper:
https://www.researchgate.net/publication/266404905_La_d_etermination_des_points_isol_es_et_de_la_dimension_d'une_vari_et_e_alg_ebrique_peut_se_faire_en_temps_polynomial?fbclid=IwAR0Ovgg0toTcuexnNIyCvETd1tiJvgIYqoJuEewuz8xxcty_Qof6_ftJxuI