5

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!

user238194
  • 681

2 Answers2

3

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 $g_2 \notin \langle x_{4}x_{5} - x_{1}^{2}x_{2}^{3}x_{3}, x_{1}^{8}x_{3} - x_{4}^{4}, x_{5}^{4} - x_{2}^{12}x_{3}^{3} \rangle$.

Viktor Vaughn
  • 19,278
0

The following items illustrate an algorithm to compute saturation of ideals. Let $I\subset S := k[x_1,\dots,x_n]$ be an ideal and fix $f\in S$. Then the saturation of $I$ w.r.t. $f$ is the set $$I:f^\infty=\{g\in S|f^m g\in I \text{ for some } m>0\}.$$

  • $I:f^\infty$ is an ideal.
  • We have the ascending chain of ideals $I:f \subset I:f^2\subset I:f^3 \subset \dots.$
  • There is an integer $N$ for which $I:f^N=I:f^{N+1}=I:f^\infty$.
  • $I:f^\infty=I:f^m$ iff $I:f^m=I:f^{m+1}$.
  • $I:f=\frac{1}{f}(I\cap \langle f \rangle )$.

Let $L\subset \mathbb{Z}^m$ be a sublattice and $\ell^1, \dots, \ell^r$ be a basis for $L$. Then the lattice ideal $I_L$ can be computed as the saturation $$I_L = \langle x^{\ell^i_{+}}-x^{\ell^i_{-}}|i=1,\dots,r\rangle:\langle x_1\cdots x_m\rangle^\infty,$$ where $\ell_{+}=\sum_{\ell_i>0}\ell_{i}e_i$ and $\ell_{-}=-\sum_{\ell_i<0}\ell_{i} e_i$, both in $\mathbb{N}^m$.

In the given exercise we define the matrix of parametrization by

$$A:=\begin{pmatrix} 4 & 0 &0 \\ 0& 4& 0 \\ 0& 0& 4\\ 8& 0& 1\\ 0& 12& 3\\ \end{pmatrix}.$$

If $V$ is a toric affine variety given by a parametrization matrix $A\in \mathbb{Z}^{s\times r}$, then the toric ideal of $V$ is the lattice ideal $I_L$, where $L$ is the kernel of the map $\mathbb{Z}^s \rightarrow \mathbb{Z}^r; e_i \mapsto i\mbox{'th row of } A$, i.e., the basis of $L$ is the set of rows of the matrix of row-syzygies of $A$.

gap> LoadPackage( "RingsForHomalg" );
true
gap> ZZ := HomalgRingOfIntegers();
Z
gap> A := HomalgMatrix( [[4,0,0],[0,4,0],[0,0,4],[8,0,1],[0,12,3]],5,3,ZZ );
<A 5 x 3 matrix over an internal ring>
gap> L := SyzygiesOfRows(A);
<A non-zero 2 x 5 matrix over an internal ring>
gap> Display( L );
[ [  -8,   0,  -1,   4,   0 ],
  [   6,  -3,   0,  -3,   1 ] ]

Hence $I_L=\langle x_1^8 x_3 - x_4^4, x_1^6 x_5 - x_2^3 x_4^3 \rangle:(x_1 x_2 x_3 x_4 x_5)^\infty$ which can easily be computed as described above.

In Gap the toric ideal can also be computed using 4ti2

gap> LoadPackage( "4ti2Interface" );
true
gap> A := [ [ 4, 0, 0 ], [ 0, 4, 0 ], [ 0, 0, 4 ], [ 8, 0, 1 ], [ 0, 12, 3 ] ];;
gap> 4ti2Interface_groebner_matrix(A);
[ [ -8, 0, -1, 4, 0 ], [ -6, 3, 0, 3, -1 ], [ -4, 6, 1, 2, -2 ], [ -2, -3, -1, 1, 1 ], [ -2, 9, 2, 1, -3 ], [ 0, -12, -3, 0, 4 ] ]