I have very little background in algebra and algebraic geometry, so please bear with me.
I am trying to show that certain systems of trigonometric polynomial equations generally have no solution. One problem among several is to show that generally $0$ is a regular value of eigenfunctions of the flat torus Laplacian, i.e., the system $T = 0 \text{ and } \nabla T = 0$ has no solution. I prefer to take an algebraic approach because this will easily generalize to my other problems.
For example consider the case that $T$ is taken from the space $\mathcal H = \operatorname{Span}\left\{\cos x \cos y, \cos x \sin y, \sin x \cos y, \sin x \sin y\right\}$.
Here is my (failed) attempt to prove it:
- "Algebraize" the system by substituting $c_1 = \cos x, s_1 = \sin x, c_2 = \cos y, s_2 = \sin y$ and adding the equations $c_1 ^2 + s_1^2 = 1$ and $c_2^2 + s_2^2 = 1$.
- Homogenize the polynomials.
- Apply a theorem from van der Waerden, Modern Algebra vol. 2, §80 that gives an algebraic condition for the system to have a nontrivial solution in complexes.
- Find one example where it has only the trivial solution, and then this must be true for "general" $T$.
Steps 1-3 are okay, but for instance in the case of $T \in \mathcal H$ above, step 4 simply fails. The reason is that the homogeneous system looks like this: $$\cases{ \text{(linear combination of }c_1 c_2, c_1 s_2, s_1 c_2, s_1 s_2\text{)} = 0 \\ \text{(linear combination of }c_1 c_2, c_1 s_2, s_1 c_2, s_1 s_2\text{)} = 0 \\ \text{(linear combination of }c_1 c_2, c_1 s_2, s_1 c_2, s_1 s_2\text{)} = 0 \\ c_1^2 + s_1^2 = A^2 \\ c_2^2 + s_2^2 = A^2 }$$ where $A$ is a homogenizing variable, and this always has the nontrivial solution $A=c_1=s_1=0, c_2=1, s_2=\textrm i$.
How should I fix my approach? (I'd like to keep it as algebraically elementary as possible. Elimination theory is okay.)
