Principal Ideals

Let $f$ be a homogeneous polynomial and $I$ be the principal ideal generated by $f$.

1. What are possible reduced Groebner bases of $I$? Describe all initial ideals of $f$.
2. Show that the Groebner fan of $I$ coincides with the normal fan of $f$.
3. Show that $\mathcal{T}(I) = \mathcal{T}(f)$.
4. Describe the Groebner fan and the tropical variety of $\langle x^3 + y^3 + z^3 + xyz\rangle$. (What are the facets? dimension? lineality?)

### Solution:

1) The only reduced Gröbner basis of $I$ is the set $\{f\}$ because every element of $I$ is a multiple of $f$ and the initial form of a product is the product of the initial forms (with respect to any term order). Therefore, the set of initial ideals is in one-to-one correspondence to the faces of the Newton polytope of $f$: The initial ideals are generated by the initial forms of $f$ with respect to the weight order under consideration. The initial form is defined as

(1)
\begin{align} \operatorname{in}_\omega(f)=\sum_a c_ax^a \end{align}

where $a$ ranges over $\{a\in\operatorname{supp}(f)\colon \forall\; b\in\operatorname{supp}(f)\; \omega\cdot a \geq \omega\cdot b\}$. This set is by definition a face of $\operatorname{New}(f)$. The vertices of $\operatorname{New}(f)$ correspond to the monomial ideals.

2) The Gröbner fan of $I$ is the set of all weight vectors such that the initial ideal with respect to the corresponding weight order does not contain a monomial. Since the initial ideal is also principal, this is the same as saying, that it is not a monomial ideal. Therefore, the weights in question correspond to linear forms exposing a face of $\operatorname{New}(f)$ that is not a vertex. This gives the correspondence between the Gröbner fan of $I$ and the cones in the normal fan of $\operatorname{New}(f)$ that are not full-dimensional (i.e. the normal cones to faces that are not vertices).

3) This follows form the fact that a principal ideal contains a monomial if and only if it is a monomial ideal. (Recall that the tropical variety of an ideal is defined as the set of all weight vectors $\omega\in\mathbb{R}^n$ such that $\operatorname{in}_\omega(I)$ contains no monomial and the tropical hypersurface of $f$ is defined as all weight vectors such that $\operatorname{in}_\omega(f)$ is not a monomial).

4) The Newton polytope of the given polynomial is a scalar multiple of the two-dimensional simplex in $\mathbb{R}^3$, namely

(2)
\begin{align} \operatorname{conv}\{(3,0,0),(0,3,0),(0,0,3),(1,1,1)=3\Delta_2\subset\mathbb{R^3}. \end{align}

It looks like a triangle in the affine plane defined by the equation $x_1+x_2+x_3=1$. Therefore the normal fan of this polytope contains a linear space, namely the line spanned by $(1,1,1)$.