Finding generators of toric ideals

Question

Consider the affine toric variety $V \subset k^{5}$ parametrized by $$\Phi(s,t,u) = (s^{4},t^{4},u^{4},s^{8}u,t^{12}u^{3}) \in k^{5}$$ where k is an algebraically closed field of characteristic 2. This is problem 1.1.8 from Cox, Little, and Schenck, but my question regards a more general notion: How exactly does one determine the generators of the toric ideals. Just by looking at the parametrization, I can deduce the generators $x_{4}x_{5} - x_{1}^{2}x_{2}^{3}x_{3}, x_{1}^{8}x_{3} - x_{4}^{4}$, and $x_{5}^{4} - x_{2}^{12}x_{3}^{3}$

Would these then generate the toric ideal? In general, is there a way to determine whether the toric ideal is the correct one other than just looking at the relations?

Thanks!

Answer 1

If you haven't already, you should consult Cox, Little, and O'Shea's Ideals, Varieties, and Algorithms: they treat this exact problem in Chapter 3 Elimination, particularly in $\S3$ Implicitization. The main result is the following theorem.

Theorem 1 (Polynomial Implicitization) If $k$ is an infinite field, let $F: k^m \to k^n$ be the function determined by the polynomial parametrization \begin{align*} x_1 &= f_1(t_1, \ldots, t_m)\\ &\ \, \vdots\\ x_n &= f_n(t_1, \ldots, t_m) \, . \end{align*} Let $I = \langle x_1 - f_1, \ldots, x_n - f_n \rangle$ and let $I_m = I \cap k[x_1, \ldots, x_n]$ be the $m^\text{th}$ elimination ideal. Then $\mathbb{V}(I_m)$ is the smallest variety in $k^n$ containing $F(k^m)$.

For your particular problem, I used Sage to compute a Gröbner basis.

R.<s,t,u,x1,x2,x3,x4,x5> = PolynomialRing(QQ,8, order = 'lex')
I_gens = [x1 - s^4, x2 - t^4, x3 - u^4, x4 - s^8*u, x5 - t^12*u^3]
I = ideal(I_gens)
basis = I.groebner_basis()

By another theorem, the generators contained in this basis only involving the variables $x_1, \ldots, x_5$ form a Gröbner basis for the elimination ideal. In this case, they are $$ \{x_1^8 x_3 - x_4^4, x_1^6 x_5 - x_2^3 x_4^3, x_1^4 x_5^2 - x_2^6 x_3 x_4^2, x_1^2 x_2^3 x_3 - x_4 x_5, x_1^2 x_5^3 - x_2^9 x_3^2 x_4, x_2^{12} x_3^3 - x_5^4\}. $$ In particular, this shows that the polynomials you found do not generate $\mathbb{I}(V)$, since for instance the second generator $g_2 = x_1^6 x_5 - x_2^3 x_4^3 \in \mathbb{I}(V)$ but $g_2 \notin \langle x_{4}x_{5} - x_{1}^{2}x_{2}^{3}x_{3}, x_{1}^{8}x_{3} - x_{4}^{4}, x_{5}^{4} - x_{2}^{12}x_{3}^{3} \rangle$.