I'd like to minimize the following polynomial in 6 variables $h_0,h_1,g_0,g_1,g_2,g_3$:
$$ g_3^2\cdot h_0^3\cdot h_1^3 - g_2\cdot g_3\cdot h_0^2\cdot h_1^4 + g_1\cdot g_3\cdot h_0\cdot h_1^5 - g_0\cdot g_3\cdot h_1^6 - 27\cdot g_3^3\cdot h_0^4 + 36\cdot g_2\cdot g_3^2\cdot h_0^3\cdot h_1 - 8\cdot g_2^2\cdot g_3\cdot h_0^2\cdot h_1^2 - 30\cdot g_1\cdot g_3^2\cdot h_0^2\cdot h_1^2 + 8\cdot g_1\cdot g_2\cdot g_3\cdot h_0\cdot h_1^3 + 36\cdot g_0\cdot g_3^2\cdot h_0\cdot h_1^3 + g_1^2\cdot g_3\cdot h_1^4 - 12\cdot g_0\cdot g_2\cdot g_3\cdot h_1^4 - 16\cdot g_2^3\cdot g_3\cdot h_0^2 + 72\cdot g_1\cdot g_2\cdot g_3^2\cdot h_0^2 - 216\cdot g_0\cdot g_3^3\cdot h_0^2 + 16\cdot g_1\cdot g_2^2\cdot g_3\cdot h_0\cdot h_1 - 96\cdot g_1^2\cdot g_3^2\cdot h_0\cdot h_1 + 144\cdot g_0\cdot g_2\cdot g_3^2\cdot h_0\cdot h_1 + 8\cdot g_1^2\cdot g_2\cdot g_3\cdot h_1^2 - 48\cdot g_0\cdot g_2^2\cdot g_3\cdot h_1^2 + 72\cdot g_0\cdot g_1\cdot g_3^2\cdot h_1^2 + 16\cdot g_1^2\cdot g_2^2\cdot g_3 - 64\cdot g_0\cdot g_2^3\cdot g_3 - 64\cdot g_1^3\cdot g_3^2 + 288\cdot g_0\cdot g_1\cdot g_2\cdot g_3^2 - 432\cdot g_0^2\cdot g_3^3 $$
Yet not over $\mathbb R^6$, but over $\mathbb Z^6$ (or just the grid $([-4;5]\cap\mathbb Z)^6$).
My problem is, that visiting all the points of the grid takes, in my example $|[-4;5]|^6=10^6$ steps, which is too much for me -- I'd like to expand the problem to more variables later.
And I want to exclude some trivial solutions like $(0,0,0,0,0,0)$. To be precise: One of the $h_i$ and two of the $g_i$ must be nonzero.
Do you have an idea how I can use the fact that the polynomial is continuous and smooth?
Kind regards,