A package for computations with classical resultants

TL;DR

This paper introduces the Resultants package in Macaulay2, enabling efficient computation of multivariate resultants, discriminants, and Chow forms, supported by algorithmic background and illustrative examples.

Contribution

The paper presents a new Macaulay2 package that implements algorithms for computing classical algebraic resultants and related invariants.

Findings

Provides effective algorithms for multivariate resultants

Includes implementation details and usage examples

Enhances computational tools in algebraic geometry

Abstract

We present the Macaulay2 package Resultants, which provides commands for the effective computation of multivariate resultants, discriminants, and Chow forms. We provide some background for the algorithms implemented and show, with a few examples, how the package works.

Equations

$$F_0=0,\dots ,F_n=0$$

$$\mathrm{Res}(F_{\sigma (0)},\dots ,F_{\sigma (n)})=\mathrm{sign}(\sigma {)}^{d_0\cdots d_n}\mathrm{Res}(F_0,\dots ,F_n).$$

$$\mathrm{Res}(F_0,\dots ,F_j,\dots ,F_n)=\mathrm{Res}(F_0,\dots ,F_j^{\prime },\dots ,F_n)\mathrm{Res}(F_0,\dots ,F_j^{\prime \prime },\dots ,F_n).$$

$$\mathrm{Res}(F_0(Ax),\dots ,F_n(Ax))={\det (A)}^{d_0\cdots d_n}\mathrm{Res}(F_0(x),\dots ,F_n(x)),$$

$$\mathrm{Res}(F_0,\dots ,F_j+\sum _{i\ne j}{H}_i{F}_i,\dots ,F_n)=\mathrm{Res}(F_0,\dots ,F_j,\dots ,F_n).$$

$$W:=\{((c_{i,\alpha }),p)\in {\mathbb{A}}^M\times {\mathbb{P}}^n:p\in V(\sum _{|\alpha |=d_0}{c}_{0,\alpha }{x}^{\alpha },\dots ,\sum _{|\alpha |=d_n}{c}_{n,\alpha }{x}^{\alpha })\}.$$

$$\mathrm{Res}(F_0,\dots ,F_n)=\mathrm{Res}(\overline{F_0},\dots ,\overline{F_{n-1}}{)}^{d_n}\det (m_{f_n}:A\to A),$$

$$\det (m_{f_n}:A\to A)=\prod _{p\in V}{f}_n(p{)}^{{\mathrm{mult}}_p(V)},$$

$$S_i:=\{x^{\alpha }:|\alpha |=\delta ,\min \{j:x_j^{d_j}|x^{\alpha }\}=i\},\text{mbox}fori=0,\dots n.$$

$$x^{\alpha }/x_i^{d_i}{F}_i,\text{mbox}fori=0,\dots n\text{mbox}and{x}^{\alpha }\in S_i.$$

$$\det (\mathbb{D}(F_0,\dots ,F_n))=\mathrm{Res}(F_0,\dots ,F_n)\det ({\mathbb{D}}^{\prime }(F_0,\dots ,F_n)).$$

$$\mathrm{Res}(F_0,F_1,F_2)={\left(1^2\det \left(\begin{array}{ccc}1& 0& 0\\ 1& 1& 0\\ 0& 1& 1\end{array}\right)\right)}^4\det \left(\begin{array}{cccccc}1& 0& 0& 0& 0& 0\\ 0& 2& 0& 0& 0& 0\\ 0& 0& 2& 0& 0& 0\\ 0& 1& 0& 1& 0& 0\\ 1& 0& 0& 0& 2& 0\\ 0& -1& 0& 1& 0& 2\end{array}\right)=16.$$

$$\partial F/\partial x_0=0,\dots ,\partial F/\partial x_n=0$$

$$\mathrm{Disc}(F)=c_{d,n}\mathrm{Res}(\frac{\partial F}{\partial x_0},\dots ,\frac{\partial F}{\partial x_n}),\text{mbox}where{c}_{d,n}=d^{\frac{(-1{)}^{n+1}-(d-1{)}^{n+1}}{d}}.$$

$$\mathrm{Disc}(F(Ax))={\det (A)}^{d(d-1{)}^n}\mathrm{Disc}(F(x)),$$

Full text

Introduction

The resultant characterizes the existence of nontrivial solutions for a square system of homogeneous polynomial equations as a condition on the coefficients. One of its important features is that it can be used to compute elimination ideals and to solve polynomial equations. Indeed, it provides one of the two main tools in elimination theory, along with Gröbner bases. The resultant of the system of equations given by the partial derivatives of a complex homogeneous polynomial $F$ is called (up to a constant factor) the discriminant of $F$. It characterizes the existence of singular points in the projective hypersurface $V(F)$ as a condition on the coefficients of $F$. In this special case, all polynomial equations have the same total degree. Every time the system of equations consists of $n+1$ polynomial equations of the same total degree $d$, the resultant has a further interesting property: it can be expressed as a polynomial of degree $d^{n}$ in the $(n+1)\times(n+1)$ minors of an $(n+1)\times\binom{n+d}{n}$ matrix, the coefficient matrix of the system of equations. This allows us to write down a generic resultant in a more compact form. The polynomial of degree $d^{n}$ so obtained is geometrically interpreted as the Chow form of the $d$-th Veronese embedding of ${\mathbb{P}}^{n}$.

The package Resultants, included with Macaulay2 [GS17], provides commands for the explicit computation of resultants and discriminants. The main algorithm used is based on the so-called Poisson formula, which reduces the computation of the resultant of $n+1$ equations to the product of the resultant of $n$ equations with the determinant of an appropriate matrix. This algorithm requires a certain genericity condition on the input polynomials, achievable with a generic change of coordinates. The package also includes tools for working with Chow forms and more generally with tangential Chow forms.

In Section 1, from a more computational point of view, we give some background information on the general theory of resultants, discriminants, and Chow forms. In Section 2, we briefly illustrate how to use the package with the help of some examples; more detailed information and examples can be found in its documentation.

1. Overview of classical resultants

In this section, we present an overview of some classically well known facts on the theory of resultants for forms in several variables. For details and proofs, we refer mainly to [GKZ94] and [CLO05]; other references are [Jou91, Jou97, dW50, Dem12, EM99, BGW88, BJ14], and [CLO07] for the case of two bivariate polynomials.

1.1. Resultants

Suppose we are given $n+1$ homogeneous polynomials $F_{0},\ldots,F_{n}$ in $n+1$ variables $x_{0},\ldots,x_{n}$ over the complex field ${\mathbb{C}}$. For $i=0,\ldots,n$, let $d_{i}$ denote the total degree of $F_{i}$ so that we can write $F_{i}=\sum_{|\alpha|=d_{i}}c_{i,\alpha}x^{\alpha}$, where $x^{\alpha}$ denotes $x_{0}^{\alpha_{0}}\cdots x_{n}^{\alpha_{n}}$. For each pair of indices $i,\alpha$, we introduce a variable $u_{i,\alpha}$ and form the universal ring of coefficients $\mathbb{U}_{d_{0},\ldots,d_{n}}:={\mathbb{Z}}[u_{i,\alpha}:i=0,\ldots,n,|\alpha|=d_{i}]$. If $P\in\mathbb{U}_{d_{0},\ldots,d_{n}}$, we denote by $P(F_{0},\ldots,F_{n})$ the element in ${\mathbb{C}}$ obtained by replacing each variable $u_{i,\alpha}$ with the corresponding coefficient $c_{i,\alpha}$.

Theorem 1.1** ([GKZ94, CLO05]).**

If we fix positive degrees $d_{0},\ldots,d_{n}$, then there is a unique polynomial $\mathop{\rm Res}\nolimits=\mathop{\rm Res}\nolimits_{d_{0},\ldots,d_{n}}\in\mathbb{U}_{d_{0},\ldots,d_{n}}$ which has the following properties:

1. (1)

if $F_{0},\ldots,F_{n}\in{\mathbb{C}}[x_{0},\ldots,x_{n}]$ are homogeneous of degrees $d_{0},\ldots,d_{n}$, then the equations

[TABLE]

have a nontrivial solution over ${\mathbb{C}}$ (i.e., $\emptyset\neq V(F_{0},\ldots,F_{n})\subset{\mathbb{P}}^{n}_{{\mathbb{C}}}$) if and only if $\mathop{\rm Res}\nolimits(F_{0},\ldots,F_{n})=0$; 2. (2)

$\mathop{\rm Res}\nolimits$* is irreducible, even when regarded as a polynomial over ${\mathbb{C}}$;* 3. (3)

$\mathop{\rm Res}\nolimits(x_{0}^{d_{0}},\ldots,x_{n}^{d_{n}})=1$.

Definition 1.2**.**

We call $\mathop{\rm Res}\nolimits(F_{0},\ldots,F_{n})$ the resultant of $F_{0},\ldots,F_{n}$.

Remark 1.3*.*

If $A$ is any commutative ring, we define the resultant of $n+1$ homogeneous polynomials $F_{0},\ldots,F_{n}\in A[x_{0},\ldots,x_{n}]$ again as $\mathop{\rm Res}\nolimits(F_{0},\ldots,F_{n})\in A$, i.e., by specializing the coefficients of the integer polynomial $\mathop{\rm Res}\nolimits$. Thus, the formation of resultants commutes with specialization.

Example 1.4**.**

The resultant is a direct generalization of the determinant. Indeed, if $d_{0}=\cdots=d_{n}=1$, then $\mathop{\rm Res}\nolimits(F_{0},\ldots,F_{n})$ equals the determinant of the $(n+1)\times(n+1)$ coefficient matrix.

Proposition 1.5** ([Jou91, Jou97]).**

The following hold:

1. (1)

(Homogeneity)* For a fixed $j$ between [math] and $n$, $\mathop{\rm Res}\nolimits$ is homogeneous in the variables $u_{j,\alpha}$, $|\alpha|=d_{j}$, of degree $d_{0}\cdots d_{j-1}d_{j+1}\cdots d_{n}$; hence its total degree is $\sum_{j=0}^{n}d_{0}\cdots d_{j-1}d_{j+1}\cdots d_{n}$.* 2. (2)

(Simmetry)* If $\sigma$ is a permutation of $\{0,\ldots,n\}$, then*

[TABLE] 3. (3)

(Multiplicativity)* If $F_{j}=F_{j}^{\prime}F_{j}^{\prime\prime}$, then we have*

[TABLE] 4. (4)

($\mathrm{SL}(n+1)$-invariance)* For each $(n+1)\times(n+1)$ matrix $A$ over ${\mathbb{C}}$, we have*

[TABLE]

where $A\,x$ denotes the product of $A$ with the column vector $(x_{0},\ldots,x_{n})^{t}$. 5. (5)

(Elementary transf.)* If $H_{i}$ is homogeneous of degree $d_{j}-d_{i}$, then*

[TABLE]

Remark 1.6*.*

On the product ${\mathbb{A}}^{M}\times{\mathbb{P}}^{n}=\mathrm{Spec}({\mathbb{C}}[u_{i,\alpha}])\times\mathrm{Proj}({\mathbb{C}}[x_{0},\ldots,x_{n}])$, where $M=\sum_{i=0}^{n}\binom{n+d_{i}}{n}$, we have an incidence variety

[TABLE]

The first projection $\pi_{1}:W\rightarrow{\mathbb{A}}^{M}$ is birational onto its image, whereas all the fibers of the second projection $\pi_{2}:W\rightarrow{\mathbb{P}}^{n}$ are linear subspaces of dimension $M-n-1$. It follows that $W$ is a smooth irreducible variety which is birational to $\overline{\pi_{1}(W)}=\pi_{1}(W)=V(\mathop{\rm Res}\nolimits_{d_{0},\ldots,d_{n}})\subset{\mathbb{A}}^{M}$.

The following result is called Poisson formula and allows one to compute resultants inductively.

Theorem 1.7** ([Jou91]; see also [CLO05]).**

Let $f_{i}(x_{0},\ldots,x_{n-1}):=F_{i}(x_{0},\ldots,x_{n-1},1)$ and $\overline{F_{i}}(x_{0},\ldots,x_{n-1}):=F_{i}(x_{0},\ldots,x_{n-1},0)$. If $\mathop{\rm Res}\nolimits(\overline{F_{0}},\ldots,\overline{F_{n-1}})\neq 0$, then the quotient ring $A={\mathbb{C}}[x_{0},\ldots,x_{n-1}]/(f_{0},\ldots,f_{n-1})$ has dimension $d_{0}\cdots d_{n-1}$ as a vector space over ${\mathbb{C}}$, and

[TABLE]

where $m_{f_{n}}:A\rightarrow A$ is the linear map given by multiplication by $f_{n}$.

With the same hypotheses as Theorem 1.7, a monomial basis for $A$ over ${\mathbb{C}}$ (useful in the implementation) can be constructed as explained in [CLO05, Chapter 2, § 2]. Note also that we have

[TABLE]

where $V=V(f_{0},\ldots,f_{n-1})$.

We now describe the most popular way to compute resultants, which is due to Macaulay [Mac02]. Let $\delta=\sum_{i=0}^{n}d_{i}-n$ and $N=\binom{n+\delta}{n}$. We can divide the monomials $x^{\alpha}$ of total degree $\delta$ into the $n+1$ mutually disjoint sets

[TABLE]

A monomial $x^{\alpha}$ of total degree $\delta$ is called reduced if $x_{i}^{d_{i}}$ divides $x^{\alpha}$ for exactly one $i$. Consider the following $N$ homogeneous polynomials of degree $\delta$:

[TABLE]

By regarding the monomials of total degree $\delta$ as unknowns, the polynomials in (1.3) form a system of $N$ linear equations in $N$ unknowns. Let $\mathbb{D}=\mathbb{D}(F_{0},\ldots,F_{n})$ denote the coefficient matrix of this linear system, and let $\mathbb{D}^{\prime}(F_{0},\ldots,F_{n})$ denote the submatrix of $\mathbb{D}$ obtained by deleting all rows and columns corresponding to reduced monomials. The following result is called the Macaulay formula and allows one to compute the resultant as a quotient of two determinants.

Theorem 1.8** ([Mac02]; see also [Jou97, CLO05]).**

The following formula holds:

[TABLE]

In several special cases, the resultant can be expressed as a single determinant (see [GKZ94, Chapter 13, Proposition 1.6]). We also mention that besides (1.4), there are other ways to represent resultants as quotients: these include Bezoutians [EM98] and Dixon matrices [KSY94]; see also [EM99] and [CLO05, p. 110]. However, all these matrices are usually of much larger size than those involved by the Poisson formula (1.1), as shown in the following simple example (see [EM99], for a comparison between Macaulay and other resultant matrices).

Example 1.9**.**

Let $F_{0}=x^{3}+y^{2}z$, $F_{1}=xy+y^{2}+xz+yz$, $F_{2}=y^{4}+z^{4}\in{\mathbb{C}}[x,y,z]$. The Poisson formula expresses $\mathop{\rm Res}\nolimits(F_{0},F_{1},F_{2})$ as the following product of determinants:

[TABLE]

The Macaulay formula yields the same result as a quotient $\det(\mathbb{D})/\det(\mathbb{D}^{\prime})$, where $\mathbb{D}$ and $\mathbb{D}^{\prime}$ are square matrices of size $36\times 36$ and $10\times 10$, respectively.

1.2. Discriminants

Let $F=\sum_{|\alpha|=d}c_{\alpha}x^{\alpha}\in{\mathbb{C}}[x_{0},\ldots,x_{n}]$ be a homogeneous polynomial of a certain degree $d$. As above, for each index $\alpha$ we introduce a variable $u_{\alpha}$ and form the universal ring of coefficients $\mathbb{U}_{d}:={\mathbb{C}}[u_{\alpha}:|\alpha|=d]$. Then one can show that, up to sign, there is a unique polynomial $\mathop{\rm Disc}\nolimits=\mathop{\rm Disc}\nolimits_{d}\in\mathbb{U}_{d}$ which has the following properties:

1. (1)

if $F\in{\mathbb{C}}[x_{0},\ldots,x_{n}]$ is homogeneous of degrees $d$, then the equations

[TABLE]

have a nontrivial solution over ${\mathbb{C}}$ (i.e., the hypersurface defined by $F$ is singular) if and only if $\mathop{\rm Disc}\nolimits(F)=0$; 2. (2)

$\mathop{\rm Disc}\nolimits$ is irreducible, even when regarded as a polynomial over ${\mathbb{C}}$.

Proposition 1.10** ([GKZ94]).**

Up to sign, we have the formula

[TABLE]

Definition 1.11**.**

We call the polynomial defined by (1.5) the discriminant of $F$.

Proposition 1.12** ([GKZ94]).**

The following hold:

1. (1)

The polynomial $\mathop{\rm Disc}\nolimits$ is homogeneous of degree $(n+1)(d-1)^{n}$. 2. (2)

For each $(n+1)\times(n+1)$ matrix $A$ over ${\mathbb{C}}$, we have

[TABLE]

where $A\,x$ denotes the product of $A$ with the column vector $(x_{0},\ldots,x_{n})^{t}$.

Geometrically, we have the following interpretation.

Proposition 1.13** ([GKZ94]).**

The discriminant hypersurface $V(\mathop{\rm Disc}\nolimits_{d})$ in the space of forms of degree $d$ on ${\mathbb{P}}^{n}$ coincides with the dual variety of the $d$-th Veronese embedding of ${\mathbb{P}}^{n}$.

1.3. Chow forms

Let $X\subset{\mathbb{P}}^{n}$ be an irreducible subvariety of dimension $k$ and degree $d$. Consider the subvariety ${Z}(X)$ in the Grassmannian ${\mathbb{G}}(n-k-1,{\mathbb{P}}^{n})$ of all $(n-k-1)$-dimensional projective subspaces of ${\mathbb{P}}^{n}$ that intersect $X$. It turns out that ${Z}(X)$ is an irreducible hypersurface of degree $d$; thus ${Z}(X)$ is defined by the vanishing of some element $R_{X}$, unique up to a constant factor, in the homogeneous component of degree $d$ of the coordinate ring of the Grassmannian ${\mathbb{G}}(n-k-1,{\mathbb{P}}^{n})$ in the Plücker embedding. This element is called the Chow form of $X$. It is notable that $X$ can be recovered from its Chow form. See [GKZ94, Chapter 3, § 2] for details.

Consider the product ${\mathbb{P}}^{k}\times X$ as a subvariety of ${\mathbb{P}}^{(k+1)(n+1)-1}$ via the Segre embedding. Identify ${\mathbb{P}}^{(k+1)(n+1)-1}$ with the projectivization ${\mathbb{P}}(\mathrm{Mat}(k+1,n+1))$ of the space of $(k+1)\times(n+1)$ matrices and consider the natural projection $\rho:{\mathbb{P}}(\mathrm{Mat}(k+1,n+1))\dashrightarrow{\mathbb{G}}(k,n)\simeq{\mathbb{G}}(n-k-1,n)$. The following result is called the Cayley trick.

Theorem 1.14** ([GKZ94]; see also [WZ94]).**

The dual variety of ${\mathbb{P}}^{k}\times X$ coincides with the closure $\overline{\rho^{-1}(Z(X))}$, where $Z(X)\subset{\mathbb{G}}(n-k-1,n)$ is the hypersurface defined by the Chow form of $X$.

The defining polynomial of the hypersurface $\overline{\rho^{-1}(Z(X))}\subset{\mathbb{P}}(\mathrm{Mat}(k+1,n+1))$ is called $X$-resultant; it provides another way of writing the Chow form of $X$.

Now, let $F_{0},\ldots,F_{n}$ be $n+1$ generic homogeneous polynomials on ${\mathbb{P}}^{n}$ of the same degree $d>0$, and let $\mathbb{M}=\mathbb{M}(F_{0},\ldots,F_{n})$ be the $(n+1)\times N$ matrix of the coefficients of these polynomials, $N=\binom{n+d}{n}$. We consider the projection $\rho_{n,d}:{\mathbb{P}}(\mathrm{Mat}(n+1,N))\dashrightarrow{\mathbb{G}}(n,N-1)\simeq{\mathbb{G}}(N-n-2,N-1)$ defined by the maximal minors of $\mathbb{M}$.

Proposition 1.15** ([GKZ94, CLO05]).**

The hypersurface of degree $(n+1)d^{n}$ in ${\mathbb{P}}(\mathrm{Mat}(n+1,N))$ defined by the resultant $\mathop{\rm Res}\nolimits(F_{0},\ldots,F_{n})$ coincides with the closure $\overline{\rho_{n,d}^{-1}(V(R_{n,d}))}$, where $R_{n,d}$ denotes the Chow form of the $d$-th Veronese embedding of ${\mathbb{P}}^{n}$. In particular, $\mathop{\rm Res}\nolimits(F_{0},\ldots,F_{n})$ is a polynomial in the maximal minors of $\mathbb{M}$.

2. Implementation

In this section, we illustrate briefly some of the methods available in the package Resultants, included with Macaulay2 [GS17]. We refer to the package documentation (which can be viewed with viewHelp Resultants) for more details and examples.

One of the main methods is resultant, which accepts as input a list of $n+1$ homogeneous polynomials in $n+1$ variables with coefficients in some commutative ring $A$ and returns an element of $A$, the resultant of the polynomials. There are no limitations on the ring $A$ because of Remark 1.3. The algorithms implemented are the Poisson formula (Theorem 1.7) and the Macaulay formula (Theorem 1.8). The former is used by default since it is typically faster, while for the latter one has to set the Algorithm option: resultant(...,Algorithm=>"Macaulay"). The method can also be configured to involve interpolation of multivariate polynomials (see [CM93]), i.e., it can reconstruct the polynomial resultant from its values at a sufficiently large number of points, which in turn are evaluated using the same formulas. The main derived method is discriminant, which applies the formula (1.5) to compute discriminants of homogeneous polynomials.

Example 2.1**.**

In the following code, we take two forms $F,G$ of degree $6$ on ${\mathbb{P}}^{3}$. We first verify that $\mathop{\rm Disc}\nolimits(F)=0$ and $\mathop{\rm Disc}\nolimits(G)\neq 0$ and then we compute the intersection of the pencil generated by $F$ and $G$ with the discriminant hypersurface in the space of forms of degree $6$ on ${\mathbb{P}}^{3}$, which is a hypersurface of degree $500$ in ${\mathbb{P}}^{83}$. (The algorithm behind these calculations is the Poisson formula; this is one of the cases where the Macaulay formula is much slower).

Macaulay2, version 1.10with packages: ConwayPolynomials, Elimination, IntegralClosure, InverseSystems,               LLLBases, PrimaryDecomposition, ReesAlgebra, TangentConei1 : loadPackage "Resultants";i2 : ZZ[w,x,y,z]; (F,G) = (w^6+x^6+y^6+wxy^4,w^6+x^6+y^6+z^6)       6    6        4    6   6    6    6    6o3 = (w  + x  + wxy  + y , w  + x  + y  + z )o3 : Sequencei4 : time discriminant F     -- used 0.0179806 secondso4 = 0i5 : time discriminant G     -- used 0.0310744 secondso5 = 140570811483169199470638017932788358544282187717397844656324826769552160278476332564065021451208552366768116974888824357602177140783996641050196723813387482285763888016904232984135762316136175977862452217324448345919411204360245828922074151228959163773714466361681597648097658753070739833449997864683601657856i6 : R := ZZ[t,u][w,x,y,z]; pencil = tsub(F,R) + usub(G,R)             6           6          4           6      6o7 = (t + u)w  + (t + u)x  + twxy  + (t + u)y  + uzo7 : ZZ[t, u][w, x, y, z]i8 : time D = discriminant pencil     -- used 7.05101 seconds                                   375 125                                 374 126 ...     11918167904272470982401...000t   u    + 44811489377450403137211...000t   u    ...o8 : ZZ[t, u]i9 : factor D        125       195    3      2         2      3 30    3      2         2      3 30o9 = (u)   (t + u)   (25t  + 81t u + 81tu  + 27u )  (29t  + 81t u + 81tu  + 27u )  (1845309860334485435604513007620143382090608492211798740863140403531458335493678485869019666668055428407222803144055042891867966935429959336227999512218285981355846846846364626801397625813957058058834010980828766582924640256)o9 : Expression of class Product In particular, we deduce that the pencil $\langle F,G\rangle$ intersects the discriminant hypersurface in $F$ with multiplicity $125$, in $F-G$ with multiplicity $195$, and in other six distinct points with multiplicity $30$.

The package also provides methods for working with Chow forms and more generally tangential Chow forms of projective varieties (see [GKZ94, p. 104] and [GM86]). In the following example, we apply some of these methods.

Example 2.2**.**

Take $C\subset{\mathbb{P}}^{3}$ to be the twisted cubic curve.

i10 : C = kernel veronese(1,3)              2                       2o10 = ideal (x  - x x , x x  - x x , x  - x x )              2    1 3   1 2    0 3   1    0 2o10 : Ideal of QQ[x , x , x , x ]                   0   1   2   3 The Chow form of $C$ in ${\mathbb{G}}(1,3)$ can be obtained as follows:

i11 : w = chowForm C       3                         2      2o11 = x    - x   x   x    + x   x    + x   x    - 2x   x   x    - x   x   x       1,2    0,2 1,2 1,3    0,1 1,3    0,2 2,3     0,1 1,2 2,3    0,1 0,3 2,3      QQ[x   , x   , x   , x   , x   , x   ]          0,1   0,2   1,2   0,3   1,3   2,3o11 : --------------------------------------          x   x    - x   x    + x   x           1,2 0,3    0,2 1,3    0,1 2,3 We can recover $C$ from its Chow form by taking the so-called Chow equations ([GKZ94, p. 102], [Cat92]).

i12 : C == saturate chowEquations wo12 = true The $X$-resultant of $C$ can be obtained applying first the duality isomorphism ${\mathbb{G}}(1,{\mathbb{P}}^{3})={\mathbb{G}}(1,{{\mathbb{P}}^{3}}^{\ast})$ and then passing from the Plücker to the Stiefel coordinates.

i13 : w’ = dualize w       3                         2      2o13 = x    - x   x   x    + x   x    + x   x    - x   x   x    - 2x   x   x       0,3    0,2 0,3 1,3    0,1 1,3    0,2 2,3    0,1 1,2 2,3     0,1 0,3 2,3      QQ[x   , x   , x   , x   , x   , x   ]          0,1   0,2   1,2   0,3   1,3   2,3o13 : --------------------------------------          x   x    - x   x    + x   x           1,2 0,3    0,2 1,3    0,1 2,3i14 : fromPluckerToStiefel w’         3   3          2   2              2       2          2   3     ...o14 = - x   x    + x   x   x   x    - x   x   x   x    + x   x   x    - ...         0,3 1,0    0,2 0,3 1,0 1,1    0,1 0,3 1,0 1,1    0,0 0,3 1,1   ...o14 : QQ[x   , x   , x   , x   , x   , x   , x   , x   ]          0,0   0,1   0,2   0,3   1,0   1,1   1,2   1,3 The method cayleyTrick returns a pair consisting of the defining ideal of ${\mathbb{P}}^{1}\times C\subset{\mathbb{P}}^{7}\simeq{\mathbb{P}}(\mathrm{Mat}(2,4))$ and the $X$-resultant of $C$, considered as a hypersurface $Z\subset{\mathbb{P}}(\mathrm{Mat}(2,4))$. Theorem 1.14 ensures that $Z$ is the dual variety of ${\mathbb{P}}^{1}\times C$. We can check this using the method dualVariety.

i15 : (P1xC,Z) = cayleyTrick C;i16 : dualVariety(P1xC) == Zo16 = true

Some overlapping packages

There are two further packages related to resultant computations, which are included in Macaulay2: Elimination, by M. E. Stillman (see [Sti05]); and EliminationMatrices, by N. Botbol, L. Busé and M. Dubinsky (see [BBD12]). The former contains functions to compute Sylvester resultants. The latter can compute different resultant matrices; in particular, it contains an implementation of the Macaulay formula.

A further package for working with Chow forms is Coisotropy, by K. Kohn (see [Koh16]), which in particular, contains a useful function to compute the degrees of all tangential Chow forms of a given projective variety.

For all these overlapping functions, it does not seem easy to rank implementations in terms of efficiency because this generally depends on the problem. They also differ in how they handle input and output. For instance, the discriminant of a binary form computed using the package Elimination lies again in the same ring, rather than in the ring of coefficients, and the Chow form of a projective variety computed using Coisotropy lies in a polynomial ring, rather than in a quotient ring.