Classical Invariant Theory

This module lists classical invariants and covariants of homogeneous polynomials (also called algebraic forms) under the action of the special linear group. That is, we are dealing with polynomials of degree \(d\) in \(n\) variables. The special linear group \(SL(n,\CC)\) acts on the variables \((x_1,\dots, x_n)\) linearly,

\[(x_1,\dots, x_n)^t \to A (x_1,\dots, x_n)^t ,\qquad A \in SL(n,\CC)\]

The linear action on the variables transforms a polynomial \(p\) generally into a different polynomial \(gp\). We can think of it as an action on the space of coefficients in \(p\). An invariant is a polynomial in the coefficients that is invariant under this action. A covariant is a polynomial in the coefficients and the variables \((x_1,\dots, x_n)\) that is invariant under the combined action.

For example, the binary quadratic \(p(x,y) = a x^2 + b x y + c y^2\) has as its invariant the discriminant \(\mathop{disc}(p) = b^2 - 4 a c\). This means that for any \(SL(2,\CC)\) coordinate change

\[\begin{split}\begin{pmatrix} x' \\ y' \end{pmatrix} = \begin{pmatrix} \alpha & \beta \\ \gamma & \delta \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} \qquad \alpha\delta-\beta\gamma=1\end{split}\]

the discriminant is invariant, \(\mathop{disc}\big(p(x',y')\big) = \mathop{disc}\big(p(x,y)\big)\).

To use this module, you should use the factory object invariant_theory. For example, take the quartic:

sage: R.<x,y> = QQ[]
sage: q = x^4 + y^4
sage: quartic = invariant_theory.binary_quartic(q);  quartic
Binary quartic with coefficients (1, 0, 0, 0, 1)

One invariant of a quartic is known as the Eisenstein D-invariant. Since it is an invariant, it is a polynomial in the coefficients (which are integers in this example):

sage: quartic.EisensteinD()
1

One example of a covariant of a quartic is the so-called g-covariant (actually, the Hessian). As with all covariants, it is a polynomial in \(x\), \(y\) and the coefficients:

sage: quartic.g_covariant()
-x^2*y^2

As usual, use tab completion and the online help to discover the implemented invariants and covariants.

In general, the variables of the defining polynomial cannot be guessed. For example, the zero polynomial can be thought of as a homogeneous polynomial of any degree. Also, since we also want to allow polynomial coefficients we cannot just take all variables of the polynomial ring as the variables of the form. This is why you will have to specify the variables explicitly if there is any potential ambiguity. For example:

sage: invariant_theory.binary_quartic(R.zero(), [x,y])
Binary quartic with coefficients (0, 0, 0, 0, 0)

sage: invariant_theory.binary_quartic(x^4, [x,y])
Binary quartic with coefficients (0, 0, 0, 0, 1)

sage: R.<x,y,t> = QQ[]
sage: invariant_theory.binary_quartic(x^4 + y^4 + t*x^2*y^2, [x,y])
Binary quartic with coefficients (1, 0, t, 0, 1)

Finally, it is often convenient to use inhomogeneous polynomials where it is understood that one wants to homogenize them. This is also supported, just define the form with an inhomogeneous polynomial and specify one less variable:

sage: R.<x,t> = QQ[]
sage: invariant_theory.binary_quartic(x^4 + 1 + t*x^2, [x])
Binary quartic with coefficients (1, 0, t, 0, 1)

REFERENCES:

[WpInvariantTheory]http://en.wikipedia.org/wiki/Glossary_of_invariant_theory
class sage.rings.invariant_theory.AlgebraicForm(n, d, polynomial, *args, **kwds)

Bases: sage.rings.invariant_theory.FormsBase

The base class of algebraic forms (i.e. homogeneous polynomials).

You should only instantiate the derived classes of this base class.

Derived classes must implement coeffs() and scaled_coeffs()

INPUT:

  • n – The number of variables.
  • d – The degree of the polynomial.
  • polynomial – The polynomial.
  • *args – The variables, as a single list/tuple, multiple arguments, or None to use all variables of the polynomial.

Derived classes must implement the same arguments for the constructor.

EXAMPLES:

sage: from sage.rings.invariant_theory import AlgebraicForm
sage: R.<x,y> = QQ[]
sage: p = x^2 + y^2
sage: AlgebraicForm(2, 2, p).variables()
(x, y)
sage: AlgebraicForm(2, 2, p, None).variables()
(x, y)
sage: AlgebraicForm(3, 2, p).variables()
(x, y, None)
sage: AlgebraicForm(3, 2, p, None).variables()
(x, y, None)

sage: from sage.rings.invariant_theory import AlgebraicForm
sage: R.<x,y,s,t> = QQ[]
sage: p = s*x^2 + t*y^2
sage: AlgebraicForm(2, 2, p, [x,y]).variables()
(x, y)
sage: AlgebraicForm(2, 2, p, x,y).variables()
(x, y)

sage: AlgebraicForm(3, 2, p, [x,y,None]).variables()
(x, y, None)
sage: AlgebraicForm(3, 2, p, x,y,None).variables()
(x, y, None)

sage: AlgebraicForm(2, 1, p, [x,y]).variables()
Traceback (most recent call last):
...
ValueError: Polynomial is of the wrong degree.

sage: AlgebraicForm(2, 2, x^2+y, [x,y]).variables()
Traceback (most recent call last):
...
ValueError: Polynomial is not homogeneous.
coefficients()

Alias for coeffs().

See the documentation for coeffs() for details.

EXAMPLES:

sage: R.<a,b,c,d,e,f,g, x,y,z> = QQ[]
sage: p = a*x^2 + b*y^2 + c*z^2 + d*x*y + e*x*z + f*y*z
sage: q = invariant_theory.quadratic_form(p, x,y,z)
sage: q.coefficients()
(a, b, c, d, e, f)
sage: q.coeffs()
(a, b, c, d, e, f)
form()

Return the defining polynomial.

OUTPUT:

The polynomial used to define the algebraic form.

EXAMPLES:

sage: R.<x,y> = QQ[]
sage: quartic = invariant_theory.binary_quartic(x^4+y^4)
sage: quartic.form()
x^4 + y^4
sage: quartic.polynomial()
x^4 + y^4
homogenized(var='h')

Return form as defined by a homogeneous polynomial.

INPUT:

  • var – either a variable name, variable index or a variable (default: 'h').

OUTPUT:

The same algebraic form, but defined by a homogeneous polynomial.

EXAMPLES:

sage: T.<t> = QQ[]
sage: quadratic = invariant_theory.binary_quadratic(t^2 + 2*t + 3)
sage: quadratic 
Binary quadratic with coefficients (1, 3, 2)
sage: quadratic.homogenized()
Binary quadratic with coefficients (1, 3, 2)
sage: quadratic == quadratic.homogenized()
True
sage: quadratic.form()
t^2 + 2*t + 3
sage: quadratic.homogenized().form()
t^2 + 2*t*h + 3*h^2

sage: R.<x,y,z> = QQ[]
sage: quadratic = invariant_theory.ternary_quadratic(x^2 + 1, [x,y])
sage: quadratic.homogenized().form()
x^2 + h^2
polynomial()

Return the defining polynomial.

OUTPUT:

The polynomial used to define the algebraic form.

EXAMPLES:

sage: R.<x,y> = QQ[]
sage: quartic = invariant_theory.binary_quartic(x^4+y^4)
sage: quartic.form()
x^4 + y^4
sage: quartic.polynomial()
x^4 + y^4
transformed(g)

Return the image under a linear transformation of the variables.

INPUT:

  • g – a \(GL(n,\CC)\) matrix or a dictionary with the

    variables as keys. A matrix is used to define the linear transformation of homogeneous variables, a dictionary acts by substitution of the variables.

OUTPUT:

A new instance of a subclass of AlgebraicForm obtained by replacing the variables of the homogeneous polynomial by their image under g.

EXAMPLES:

sage: R.<x,y,z> = QQ[]
sage: cubic = invariant_theory.ternary_cubic(x^3 + 2*y^3 + 3*z^3 + 4*x*y*z)
sage: cubic.transformed({x:y, y:z, z:x}).form()
3*x^3 + y^3 + 4*x*y*z + 2*z^3
sage: cyc = matrix([[0,1,0],[0,0,1],[1,0,0]])
sage: cubic.transformed(cyc) == cubic.transformed({x:y, y:z, z:x})
True
sage: g = matrix(QQ, [[1, 0, 0], [-1, 1, -3], [-5, -5, 16]])
sage: cubic.transformed(g)
Ternary cubic with coefficients (-356, -373, 12234, -1119, 3578, -1151, 
3582, -11766, -11466, 7360)
sage: cubic.transformed(g).transformed(g.inverse()) == cubic
True
class sage.rings.invariant_theory.BinaryQuartic(n, d, polynomial, *args)

Bases: sage.rings.invariant_theory.AlgebraicForm

Invariant theory of a binary quartic.

You should use the invariant_theory factory object to construct instances of this class. See binary_quartic() for details.

TESTS:

sage: R.<a0, a1, a2, a3, a4, x0, x1> = QQ[]
sage: p = a0*x1^4 + a1*x1^3*x0 + a2*x1^2*x0^2 + a3*x1*x0^3 + a4*x0^4
sage: quartic = invariant_theory.binary_quartic(p, x0, x1)
sage: quartic._check_covariant('form')
sage: quartic._check_covariant('EisensteinD', invariant=True)
sage: quartic._check_covariant('EisensteinE', invariant=True)
sage: quartic._check_covariant('g_covariant')
sage: quartic._check_covariant('h_covariant')
sage: TestSuite(quartic).run()
EisensteinD()

One of the Eisenstein invariants of a binary quartic.

OUTPUT:

The Eisenstein D-invariant of the quartic.

\[\begin{split}f(x) = a_0 x_1^4 + 4 a_1 x_0 x_1^3 + 6 a_2 x_0^2 x_1^2 + 4 a_3 x_0^3 x_1 + a_4 x_0^4 \\ \Rightarrow D(f) = a_0 a_4+3 a_2^2-4 a_1 a_3\end{split}\]

EXAMPLES:

sage: R.<a0, a1, a2, a3, a4, x0, x1> = QQ[]
sage: f = a0*x1^4+4*a1*x0*x1^3+6*a2*x0^2*x1^2+4*a3*x0^3*x1+a4*x0^4
sage: inv = invariant_theory.binary_quartic(f, x0, x1)
sage: inv.EisensteinD()
3*a2^2 - 4*a1*a3 + a0*a4
EisensteinE()

One of the Eisenstein invariants of a binary quartic.

OUTPUT:

The Eisenstein E-invariant of the quartic.

\[\begin{split}f(x) = a_0 x_1^4 + 4 a_1 x_0 x_1^3 + 6 a_2 x_0^2 x_1^2 + 4 a_3 x_0^3 x_1 + a_4 x_0^4 \\ \Rightarrow E(f) = a_0 a_3^2 +a_1^2 a_4 -a_0 a_2 a_4 -2 a_1 a_2 a_3 + a_2^3\end{split}\]

EXAMPLES:

sage: R.<a0, a1, a2, a3, a4, x0, x1> = QQ[]
sage: f = a0*x1^4+4*a1*x0*x1^3+6*a2*x0^2*x1^2+4*a3*x0^3*x1+a4*x0^4
sage: inv = invariant_theory.binary_quartic(f, x0, x1)
sage: inv.EisensteinE()
a2^3 - 2*a1*a2*a3 + a0*a3^2 + a1^2*a4 - a0*a2*a4
coeffs()

The coefficients of a binary quartic.

Given

\[f(x) = a_0 x_1^4 + a_1 x_0 x_1^3 + a_2 x_0^2 x_1^2 + a_3 x_0^3 x_1 + a_4 x_0^4\]

this function returns \(a = (a_0, a_1, a_2, a_3, a_4)\)

EXAMPLES:

sage: R.<a0, a1, a2, a3, a4, x0, x1> = QQ[]
sage: p = a0*x1^4 + a1*x1^3*x0 + a2*x1^2*x0^2 + a3*x1*x0^3 + a4*x0^4
sage: quartic = invariant_theory.binary_quartic(p, x0, x1)
sage: quartic.coeffs()
(a0, a1, a2, a3, a4)

sage: R.<a0, a1, a2, a3, a4, x> = QQ[]
sage: p = a0 + a1*x + a2*x^2 + a3*x^3 + a4*x^4
sage: quartic = invariant_theory.binary_quartic(p, x)
sage: quartic.coeffs()
(a0, a1, a2, a3, a4)
g_covariant()

The g-covariant of a binary quartic.

OUTPUT:

The g-covariant of the quartic.

\[\begin{split}f(x) = a_0 x_1^4 + 4 a_1 x_0 x_1^3 + 6 a_2 x_0^2 x_1^2 + 4 a_3 x_0^3 x_1 + a_4 x_0^4 \\ \Rightarrow D(f) = \frac{1}{144} \begin{pmatrix} \frac{\partial^2 f}{\partial x \partial x} \end{pmatrix}\end{split}\]

EXAMPLES:

sage: R.<a0, a1, a2, a3, a4, x, y> = QQ[]
sage: p = a0*x^4+4*a1*x^3*y+6*a2*x^2*y^2+4*a3*x*y^3+a4*y^4
sage: inv = invariant_theory.binary_quartic(p, x, y)
sage: g = inv.g_covariant();  g
a1^2*x^4 - a0*a2*x^4 + 2*a1*a2*x^3*y - 2*a0*a3*x^3*y + 3*a2^2*x^2*y^2
- 2*a1*a3*x^2*y^2 - a0*a4*x^2*y^2 + 2*a2*a3*x*y^3
- 2*a1*a4*x*y^3 + a3^2*y^4 - a2*a4*y^4

sage: inv_inhomogeneous = invariant_theory.binary_quartic(p.subs(y=1), x)
sage: inv_inhomogeneous.g_covariant()
a1^2*x^4 - a0*a2*x^4 + 2*a1*a2*x^3 - 2*a0*a3*x^3 + 3*a2^2*x^2
- 2*a1*a3*x^2 - a0*a4*x^2 + 2*a2*a3*x - 2*a1*a4*x + a3^2 - a2*a4

sage: g == 1/144 * (p.derivative(x,y)^2 - p.derivative(x,x)*p.derivative(y,y))
True
h_covariant()

The h-covariant of a binary quartic.

OUTPUT:

The h-covariant of the quartic.

\[\begin{split}f(x) = a_0 x_1^4 + 4 a_1 x_0 x_1^3 + 6 a_2 x_0^2 x_1^2 + 4 a_3 x_0^3 x_1 + a_4 x_0^4 \\ \Rightarrow D(f) = \frac{1}{144} \begin{pmatrix} \frac{\partial^2 f}{\partial x \partial x} \end{pmatrix}\end{split}\]

EXAMPLES:

sage: R.<a0, a1, a2, a3, a4, x, y> = QQ[]
sage: p = a0*x^4+4*a1*x^3*y+6*a2*x^2*y^2+4*a3*x*y^3+a4*y^4
sage: inv = invariant_theory.binary_quartic(p, x, y)
sage: h = inv.h_covariant();   h
-2*a1^3*x^6 + 3*a0*a1*a2*x^6 - a0^2*a3*x^6 - 6*a1^2*a2*x^5*y + 9*a0*a2^2*x^5*y
- 2*a0*a1*a3*x^5*y - a0^2*a4*x^5*y - 10*a1^2*a3*x^4*y^2 + 15*a0*a2*a3*x^4*y^2
- 5*a0*a1*a4*x^4*y^2 + 10*a0*a3^2*x^3*y^3 - 10*a1^2*a4*x^3*y^3
+ 10*a1*a3^2*x^2*y^4 - 15*a1*a2*a4*x^2*y^4 + 5*a0*a3*a4*x^2*y^4
+ 6*a2*a3^2*x*y^5 - 9*a2^2*a4*x*y^5 + 2*a1*a3*a4*x*y^5 + a0*a4^2*x*y^5
+ 2*a3^3*y^6 - 3*a2*a3*a4*y^6 + a1*a4^2*y^6

sage: inv_inhomogeneous = invariant_theory.binary_quartic(p.subs(y=1), x)
sage: inv_inhomogeneous.h_covariant()
-2*a1^3*x^6 + 3*a0*a1*a2*x^6 - a0^2*a3*x^6 - 6*a1^2*a2*x^5 + 9*a0*a2^2*x^5
- 2*a0*a1*a3*x^5 - a0^2*a4*x^5 - 10*a1^2*a3*x^4 + 15*a0*a2*a3*x^4
- 5*a0*a1*a4*x^4 + 10*a0*a3^2*x^3 - 10*a1^2*a4*x^3 + 10*a1*a3^2*x^2
- 15*a1*a2*a4*x^2 + 5*a0*a3*a4*x^2 + 6*a2*a3^2*x - 9*a2^2*a4*x
+ 2*a1*a3*a4*x + a0*a4^2*x + 2*a3^3 - 3*a2*a3*a4 + a1*a4^2

sage: g = inv.g_covariant()
sage: h == 1/8 * (p.derivative(x)*g.derivative(y)-p.derivative(y)*g.derivative(x))
True
monomials()

List the basis monomials in the form.

OUTPUT:

A tuple of monomials. They are in the same order as coeffs().

EXAMPLES:

sage: R.<x,y> = QQ[]
sage: quartic = invariant_theory.binary_quartic(x^4+y^4)
sage: quartic.monomials()
(y^4, x*y^3, x^2*y^2, x^3*y, x^4)
scaled_coeffs()

The coefficients of a binary quartic.

Given

\[f(x) = a_0 x_1^4 + 4 a_1 x_0 x_1^3 + 6 a_2 x_0^2 x_1^2 + 4 a_3 x_0^3 x_1 + a_4 x_0^4\]

this function returns \(a = (a_0, a_1, a_2, a_3, a_4)\)

EXAMPLES:

sage: R.<a0, a1, a2, a3, a4, x0, x1> = QQ[]
sage: quartic = a0*x1^4 + 4*a1*x1^3*x0 + 6*a2*x1^2*x0^2 + 4*a3*x1*x0^3 + a4*x0^4
sage: inv = invariant_theory.binary_quartic(quartic, x0, x1)
sage: inv.scaled_coeffs()
(a0, a1, a2, a3, a4)

sage: R.<a0, a1, a2, a3, a4, x> = QQ[]
sage: quartic = a0 + 4*a1*x + 6*a2*x^2 + 4*a3*x^3 + a4*x^4
sage: inv = invariant_theory.binary_quartic(quartic, x)
sage: inv.scaled_coeffs()
(a0, a1, a2, a3, a4)
class sage.rings.invariant_theory.FormsBase(n, homogeneous, ring, variables)

Bases: sage.structure.sage_object.SageObject

The common base class of AlgebraicForm and SeveralAlgebraicForms.

This is an abstract base class to provide common methods. It does not make much sense to instantiate it.

TESTS:

sage: from sage.rings.invariant_theory import FormsBase
sage: FormsBase(None, None, None, None)
<class 'sage.rings.invariant_theory.FormsBase'>
is_homogeneous()

Return whether the forms were defined by homogeneous polynomials.

OUTPUT:

Boolean. Whether the user originally defined the form via homogeneous variables.

EXAMPLES:

sage: R.<x,y,t> = QQ[]
sage: quartic = invariant_theory.binary_quartic(x^4+y^4+t*x^2*y^2, [x,y])
sage: quartic.is_homogeneous()
True
sage: quartic.form()
x^2*y^2*t + x^4 + y^4

sage: R.<x,y,t> = QQ[]
sage: quartic = invariant_theory.binary_quartic(x^4+1+t*x^2, [x])
sage: quartic.is_homogeneous()
False
sage: quartic.form()
x^4 + x^2*t + 1
ring()

Return the polynomial ring.

OUTPUT:

A polynomial ring. This is where the defining polynomial(s) live. Note that the polynomials may be homogeneous or inhomogeneous, depending on how the user constructed the object.

EXAMPLES:

sage: R.<x,y,t> = QQ[]
sage: quartic = invariant_theory.binary_quartic(x^4+y^4+t*x^2*y^2, [x,y])
sage: quartic.ring()
Multivariate Polynomial Ring in x, y, t over Rational Field

sage: R.<x,y,t> = QQ[]
sage: quartic = invariant_theory.binary_quartic(x^4+1+t*x^2, [x])
sage: quartic.ring()
Multivariate Polynomial Ring in x, y, t over Rational Field
variables()

Return the variables of the form.

OUTPUT:

A tuple of variables. If inhomogeneous notation is used for the defining polynomial then the last entry will be None.

EXAMPLES:

sage: R.<x,y,t> = QQ[]
sage: quartic = invariant_theory.binary_quartic(x^4+y^4+t*x^2*y^2, [x,y])
sage: quartic.variables()
(x, y)

sage: R.<x,y,t> = QQ[]
sage: quartic = invariant_theory.binary_quartic(x^4+1+t*x^2, [x])
sage: quartic.variables()
(x, None)
class sage.rings.invariant_theory.InvariantTheoryFactory

Bases: object

Factory object for invariants of multilinear forms.

EXAMPLES:

sage: R.<x,y,z> = QQ[]
sage: invariant_theory.ternary_cubic(x^3+y^3+z^3)
Ternary cubic with coefficients (1, 1, 1, 0, 0, 0, 0, 0, 0, 0)
binary_quadratic(quadratic, *args)

Invariant theory of a quadratic in two variables.

INPUT:

  • quadratic – a quadratic form.
  • x, y – the homogeneous variables. If y is None, the quadratic is assumed to be inhomogeneous.

REFERENCES:

EXAMPLES:

sage: R.<x,y> = QQ[]
sage: invariant_theory.binary_quadratic(x^2+y^2)
Binary quadratic with coefficients (1, 1, 0)

sage: T.<t> = QQ[]
sage: invariant_theory.binary_quadratic(t^2 + 2*t + 1, [t])
Binary quadratic with coefficients (1, 1, 2)
binary_quartic(quartic, *args, **kwds)

Invariant theory of a quartic in two variables.

The algebra of invariants of a quartic form is generated by invariants \(i\), \(j\) of degrees 2, 3. This ring is naturally isomorphic to the ring of modular forms of level 1, with the two generators corresponding to the Eisenstein series \(E_4\) (see EisensteinD()) and \(E_6\) (see EisensteinE()). The algebra of covariants is generated by these two invariants together with the form \(f\) of degree 1 and order 4, the Hessian \(g\) (see g_covariant()) of degree 2 and order 4, and a covariant \(h\) (see h_covariant()) of degree 3 and order 6. They are related by a syzygy

\[j f^3 - g f^2 i + 4 g^3 + h^2 = 0\]

of degree 6 and order 12.

INPUT:

  • quartic – a quartic.
  • x, y – the homogeneous variables. If y is None, the quartic is assumed to be inhomogeneous.

REFERENCES:

EXAMPLES:

sage: R.<x,y> = QQ[]
sage: quartic = invariant_theory.binary_quartic(x^4+y^4)
sage: quartic
Binary quartic with coefficients (1, 0, 0, 0, 1)
sage: type(quartic)
<class 'sage.rings.invariant_theory.BinaryQuartic'>
inhomogeneous_quadratic_form(polynomial, *args)

Invariants of an inhomogeneous quadratic form.

INPUT:

  • polynomial – an inhomogeneous quadratic form.
  • *args – the variables as multiple arguments, or as a single list/tuple.

EXAMPLES:

sage: R.<x,y,z> = QQ[]
sage: quadratic = x^2+2*y^2+3*x*y+4*x+5*y+6
sage: inv3 = invariant_theory.inhomogeneous_quadratic_form(quadratic)
sage: type(inv3)
<class 'sage.rings.invariant_theory.TernaryQuadratic'>
sage: inv4 = invariant_theory.inhomogeneous_quadratic_form(x^2+y^2+z^2)
sage: type(inv4)
<class 'sage.rings.invariant_theory.QuadraticForm'>
quadratic_form(polynomial, *args)

Invariants of a homogeneous quadratic form.

INPUT:

  • polynomial – a homogeneous or inhomogeneous quadratic form.
  • *args – the variables as multiple arguments, or as a single list/tuple. If the last argument is None, the cubic is assumed to be inhomogeneous.

EXAMPLES:

sage: R.<x,y,z> = QQ[]
sage: quadratic = x^2+y^2+z^2
sage: inv = invariant_theory.quadratic_form(quadratic)
sage: type(inv)
<class 'sage.rings.invariant_theory.TernaryQuadratic'>

If some of the ring variables are to be treated as coefficients you need to specify the polynomial variables:

sage: R.<x,y,z, a,b> = QQ[]
sage: quadratic = a*x^2+b*y^2+z^2+2*y*z
sage: invariant_theory.quadratic_form(quadratic, x,y,z)
Ternary quadratic with coefficients (a, b, 1, 0, 0, 2)
sage: invariant_theory.quadratic_form(quadratic, [x,y,z])   # alternate syntax
Ternary quadratic with coefficients (a, b, 1, 0, 0, 2)

Inhomogeneous quadratic forms (see also inhomogeneous_quadratic_form()) can be specified by passing None as the last variable:

sage: inhom = quadratic.subs(z=1)
sage: invariant_theory.quadratic_form(inhom, x,y,None)
Ternary quadratic with coefficients (a, b, 1, 0, 0, 2)
quaternary_biquadratic(quadratic1, quadratic2, *args, **kwds)

Invariants of two quadratics in four variables.

INPUT:

  • quadratic1, quadratic2 – two polynomias. Either homogeneous quadratic in 4 homogeneous variables, or inhomogeneous quadratic in 3 variables.
  • w, x, y, z – the variables. If z is None, the quadratics are assumed to be inhomogeneous.

EXAMPLES:

sage: R.<w,x,y,z> = QQ[]
sage: q1 = w^2+x^2+y^2+z^2
sage: q2 = w*x + y*z
sage: inv = invariant_theory.quaternary_biquadratic(q1, q2)
sage: type(inv)
<class 'sage.rings.invariant_theory.TwoQuaternaryQuadratics'>

Distance between two spheres [Salmon]

sage: R.<x,y,z, a,b,c, r1,r2> = QQ[]
sage: S1 = -r1^2 + x^2 + y^2 + z^2 
sage: S2 = -r2^2 + (x-a)^2 + (y-b)^2 + (z-c)^2 
sage: inv = invariant_theory.quaternary_biquadratic(S1, S2, [x, y, z])
sage: inv.Delta_invariant()
-r1^2
sage: inv.Delta_prime_invariant()
-r2^2
sage: inv.Theta_invariant()
a^2 + b^2 + c^2 - 3*r1^2 - r2^2
sage: inv.Theta_prime_invariant()
a^2 + b^2 + c^2 - r1^2 - 3*r2^2
sage: inv.Phi_invariant()
2*a^2 + 2*b^2 + 2*c^2 - 3*r1^2 - 3*r2^2
sage: inv.J_covariant()
0
quaternary_quadratic(quadratic, *args)

Invariant theory of a quadratic in four variables.

INPUT:

  • quadratic – a quadratic form.
  • w, x, y, z – the homogeneous variables. If z is None, the quadratic is assumed to be inhomogeneous.

REFERENCES:

[WpBinaryForm]http://en.wikipedia.org/wiki/Invariant_of_a_binary_form

EXAMPLES:

sage: R.<w,x,y,z> = QQ[]
sage: invariant_theory.quaternary_quadratic(w^2+x^2+y^2+z^2)
Quaternary quadratic with coefficients (1, 1, 1, 1, 0, 0, 0, 0, 0, 0)

sage: R.<x,y,z> = QQ[]
sage: invariant_theory.quaternary_quadratic(1+x^2+y^2+z^2)
Quaternary quadratic with coefficients (1, 1, 1, 1, 0, 0, 0, 0, 0, 0)
ternary_cubic(cubic, *args, **kwds)

Invariants of a cubic in three variables.

The algebra of invariants of a ternary cubic under \(SL_3(\CC)\) is a polynomial algebra generated by two invariants \(S\) (see S_invariant()) and T (see T_invariant()) of degrees 4 and 6, called Aronhold invariants.

The ring of covariants is given as follows. The identity covariant U of a ternary cubic has degree 1 and order 3. The Hessian \(H\) (see Hessian()) is a covariant of ternary cubics of degree 3 and order 3. There is a covariant \(\Theta\) (see Theta_covariant()) of ternary cubics of degree 8 and order 6 that vanishes on points \(x\) lying on the Salmon conic of the polar of \(x\) with respect to the curve and its Hessian curve. The Brioschi covariant \(J\) (see J_covariant()) is the Jacobian of \(U\), \(\Theta\), and \(H\) of degree 12, order 9. The algebra of covariants of a ternary cubic is generated over the ring of invariants by \(U\), \(\Theta\), \(H\), and \(J\), with a relation

\[\begin{split}\begin{split} J^2 =& 4 \Theta^3 + T U^2 \Theta^2 + \Theta (-4 S^3 U^4 + 2 S T U^3 H - 72 S^2 U^2 H^2 \\ & - 18 T U H^3 + 108 S H^4) -16 S^4 U^5 H - 11 S^2 T U^4 H^2 \\ & -4 T^2 U^3 H^3 +54 S T U^2 H^4 -432 S^2 U H^5 -27 T H^6 \end{split}\end{split}\]

REFERENCES:

[WpTernaryCubic]http://en.wikipedia.org/wiki/Ternary_cubic

INPUT:

  • cubic – a homogeneous cubic in 3 homogeneous variables, or an inhomogeneous cubic in 2 variables.
  • x, y, z – the variables. If z is None, the cubic is assumed to be inhomogeneous.

EXAMPLES:

sage: R.<x,y,z> = QQ[]
sage: cubic = invariant_theory.ternary_cubic(x^3+y^3+z^3)
sage: type(cubic)
<class 'sage.rings.invariant_theory.TernaryCubic'>
ternary_quadratic(quadratic, *args, **kwds)

Invariants of a quadratic in three variables.

INPUT:

  • quadratic – a homogeneous quadratic in 3 homogeneous variables, or an inhomogeneous quadratic in 2 variables.
  • x, y, z – the variables. If z is None, the quadratic is assumed to be inhomogeneous.

REFERENCES:

EXAMPLES:

sage: R.<x,y,z> = QQ[]
sage: invariant_theory.ternary_quadratic(x^2+y^2+z^2)
Ternary quadratic with coefficients (1, 1, 1, 0, 0, 0)

sage: T.<u, v> = QQ[]
sage: invariant_theory.ternary_quadratic(1+u^2+v^2)
Ternary quadratic with coefficients (1, 1, 1, 0, 0, 0)

sage: quadratic = x^2+y^2+z^2
sage: inv = invariant_theory.ternary_quadratic(quadratic)
sage: type(inv)
<class 'sage.rings.invariant_theory.TernaryQuadratic'>
class sage.rings.invariant_theory.QuadraticForm(n, d, polynomial, *args)

Bases: sage.rings.invariant_theory.AlgebraicForm

Invariant theory of a multivariate quadratic form.

You should use the invariant_theory factory object to construct instances of this class. See quadratic_form() for details.

TESTS:

sage: R.<a,b,c,d,e,f,g, x,y,z> = QQ[]
sage: p = a*x^2 + b*y^2 + c*z^2 + d*x*y + e*x*z + f*y*z
sage: invariant_theory.quadratic_form(p, x,y,z)
Ternary quadratic with coefficients (a, b, c, d, e, f)
sage: type(_)
<class 'sage.rings.invariant_theory.TernaryQuadratic'>

sage: R.<a,b,c,d,e,f,g, x,y,z> = QQ[]
sage: p = a*x^2 + b*y^2 + c*z^2 + d*x*y + e*x*z + f*y*z
sage: invariant_theory.quadratic_form(p, x,y,z)
Ternary quadratic with coefficients (a, b, c, d, e, f)
sage: type(_)
<class 'sage.rings.invariant_theory.TernaryQuadratic'>

Since we cannot always decide whether the form is homogeneous or not based on the number of variables, you need to explicitly specify it if you want the variables to be treated as inhomogeneous:

sage: invariant_theory.inhomogeneous_quadratic_form(p.subs(z=1), x,y)
Ternary quadratic with coefficients (a, b, c, d, e, f)
as_QuadraticForm()

Convert into a QuadraticForm.

OUTPUT:

Sage has a special quadratic forms subsystem. This method converts self into this QuadraticForm representation.

EXAMPLES:

sage: R.<x,y,z> = QQ[]
sage: p = x^2+y^2+z^2+2*x*y+3*x*z
sage: quadratic = invariant_theory.ternary_quadratic(p)
sage: matrix(quadratic)
[  1   1 3/2]
[  1   1   0]
[3/2   0   1]
sage: quadratic.as_QuadraticForm()
Quadratic form in 3 variables over Multivariate Polynomial
Ring in x, y, z over Rational Field with coefficients:
[ 1/2 1 3/2 ]
[ * 1/2 0 ]
[ * * 1/2 ]
sage: _.polynomial('X,Y,Z')
X^2 + 2*X*Y + Y^2 + 3*X*Z + Z^2
coeffs()

The coefficients of a quadratic form.

Given

\[\begin{split}f(x) = \sum_{0\leq i<n} a_i x_i^2 + \sum_{0\leq j <k<n} a_{jk} x_j x_k\end{split}\]

this function returns \(a = (a_0, \dots, a_n, a_{00}, a_{01}, \dots, a_{n-1,n})\)

EXAMPLES:

sage: R.<a,b,c,d,e,f,g, x,y,z> = QQ[]
sage: p = a*x^2 + b*y^2 + c*z^2 + d*x*y + e*x*z + f*y*z
sage: inv = invariant_theory.quadratic_form(p, x,y,z); inv
Ternary quadratic with coefficients (a, b, c, d, e, f)
sage: inv.coeffs()
(a, b, c, d, e, f)
sage: inv.scaled_coeffs()
(a, b, c, 1/2*d, 1/2*e, 1/2*f)
discriminant()

Return the discriminant of the quadratic form.

Up to an overall constant factor, this is just the determinant of the defining matrix, see matrix(). For a quadratic form in \(n\) variables, the overall constant is \(2^{n-1}\) if \(n\) is odd and \((-1)^{n/2} 2^n\) if \(n\) is even.

EXAMPLES:

sage: R.<a,b,c, x,y> = QQ[]
sage: p = a*x^2+b*x*y+c*y^2
sage: quadratic = invariant_theory.quadratic_form(p, x,y)
sage: quadratic.discriminant()
b^2 - 4*a*c

sage: R.<a,b,c,d,e,f,g, x,y,z> = QQ[]
sage: p = a*x^2 + b*y^2 + c*z^2 + d*x*y + e*x*z + f*y*z
sage: quadratic = invariant_theory.quadratic_form(p, x,y,z)
sage: quadratic.discriminant()
4*a*b*c - c*d^2 - b*e^2 + d*e*f - a*f^2
dual()

Return the dual quadratic form.

OUTPUT:

A new quadratic form (with the same number of variables) defined by the adjoint matrix.

EXAMPLES:

sage: R.<a,b,c,x,y,z> = QQ[]
sage: cubic = x^2+y^2+z^2
sage: quadratic = invariant_theory.ternary_quadratic(a*x^2+b*y^2+c*z^2, [x,y,z])
sage: quadratic.form()
a*x^2 + b*y^2 + c*z^2
sage: quadratic.dual().form()
b*c*x^2 + a*c*y^2 + a*b*z^2

sage: R.<x,y,z, t> = QQ[]
sage: cubic = x^2+y^2+z^2
sage: quadratic = invariant_theory.ternary_quadratic(x^2+y^2+z^2 + t*x*y, [x,y,z])
sage: quadratic.dual()
Ternary quadratic with coefficients (1, 1, -1/4*t^2 + 1, -t, 0, 0)

sage: R.<x,y, t> = QQ[]
sage: quadratic = invariant_theory.ternary_quadratic(x^2+y^2+1 + t*x*y, [x,y])
sage: quadratic.dual()
Ternary quadratic with coefficients (1, 1, -1/4*t^2 + 1, -t, 0, 0)

TESTS:

sage: R = PolynomialRing(QQ, 'a20,a11,a02,a10,a01,a00,x,y,z', order='lex')
sage: R.inject_variables()
Defining a20, a11, a02, a10, a01, a00, x, y, z
sage: p = ( a20*x^2 + a11*x*y + a02*y^2 +
...         a10*x*z + a01*y*z + a00*z^2 )
sage: quadratic = invariant_theory.ternary_quadratic(p, x,y,z)
sage: quadratic.dual().dual().form().factor()
(1/4) * 
(a20*x^2 + a11*x*y + a02*y^2 + a10*x*z + a01*y*z + a00*z^2) *
(4*a20*a02*a00 - a20*a01^2 - a11^2*a00 + a11*a10*a01 - a02*a10^2)

sage: R.<w,x,y,z> = QQ[]
sage: q = invariant_theory.quaternary_quadratic(w^2+2*x^2+3*y^2+4*z^2+x*y+5*w*z)
sage: q.form()
w^2 + 2*x^2 + x*y + 3*y^2 + 5*w*z + 4*z^2
sage: q.dual().dual().form().factor()
(42849/256) * (w^2 + 2*x^2 + x*y + 3*y^2 + 5*w*z + 4*z^2)

sage: R.<x,y,z> = QQ[]
sage: q = invariant_theory.quaternary_quadratic(1+2*x^2+3*y^2+4*z^2+x*y+5*z)
sage: q.form()
2*x^2 + x*y + 3*y^2 + 4*z^2 + 5*z + 1
sage: q.dual().dual().form().factor()
(42849/256) * (2*x^2 + x*y + 3*y^2 + 4*z^2 + 5*z + 1)
matrix()

Return the quadratic form as a symmetric matrix

OUTPUT:

This method returns a symmetric matrix \(A\) such that the quadratic \(Q\) equals

\[Q(x,y,z,\dots) = (x,y,\dots) A (x,y,\dots)^t\]

EXAMPLES:

sage: R.<x,y,z> = QQ[]
sage: quadratic = invariant_theory.ternary_quadratic(x^2+y^2+z^2+x*y)
sage: matrix(quadratic)
[  1 1/2   0]
[1/2   1   0]
[  0   0   1]
sage: quadratic._matrix_() == matrix(quadratic)
True
monomials()

List the basis monomials in the form.

OUTPUT:

A tuple of monomials. They are in the same order as coeffs().

EXAMPLES:

sage: R.<x,y> = QQ[]
sage: quadratic = invariant_theory.quadratic_form(x^2+y^2)
sage: quadratic.monomials()
(x^2, y^2, x*y)

sage: quadratic = invariant_theory.inhomogeneous_quadratic_form(x^2+y^2)
sage: quadratic.monomials()
(x^2, y^2, 1, x*y, x, y)
scaled_coeffs()

The scaled coefficients of a quadratic form.

Given

\[\begin{split}f(x) = \sum_{0\leq i<n} a_i x_i^2 + \sum_{0\leq j <k<n} 2 a_{jk} x_j x_k\end{split}\]

this function returns \(a = (a_0, \cdots, a_n, a_{00}, a_{01}, \dots, a_{n-1,n})\)

EXAMPLES:

sage: R.<a,b,c,d,e,f,g, x,y,z> = QQ[]
sage: p = a*x^2 + b*y^2 + c*z^2 + d*x*y + e*x*z + f*y*z
sage: inv = invariant_theory.quadratic_form(p, x,y,z); inv
Ternary quadratic with coefficients (a, b, c, d, e, f)
sage: inv.coeffs()
(a, b, c, d, e, f)
sage: inv.scaled_coeffs()
(a, b, c, 1/2*d, 1/2*e, 1/2*f)
class sage.rings.invariant_theory.SeveralAlgebraicForms(forms)

Bases: sage.rings.invariant_theory.FormsBase

The base class of multiple algebraic forms (i.e. homogeneous polynomials).

You should only instantiate the derived classes of this base class.

See AlgebraicForm for the base class of a single algebraic form.

INPUT:

  • forms – a list/tuple/iterable of at least one AlgebraicForm object, all with the same number of variables. Interpreted as multiple homogeneous polynomials in a common polynomial ring.

EXAMPLES:

sage: from sage.rings.invariant_theory import AlgebraicForm, SeveralAlgebraicForms
sage: R.<x,y> = QQ[]
sage: p = AlgebraicForm(2, 2, x^2, (x,y))
sage: q = AlgebraicForm(2, 2, y^2, (x,y))
sage: pq = SeveralAlgebraicForms([p, q])
get_form(i)

Return the \(i\)-th form.

EXAMPLES:

sage: R.<x,y> = QQ[]
sage: q1 = invariant_theory.quadratic_form(x^2 + y^2)
sage: q2 = invariant_theory.quadratic_form(x*y)
sage: from sage.rings.invariant_theory import SeveralAlgebraicForms
sage: q12 = SeveralAlgebraicForms([q1, q2])
sage: q12.get_form(0) is q1
True
sage: q12.get_form(1) is q2
True
sage: q12[0] is q12.get_form(0)   # syntactic sugar
True
sage: q12[1] is q12.get_form(1)   # syntactic sugar
True
homogenized(var='h')

Return form as defined by a homogeneous polynomial.

INPUT:

  • var – either a variable name, variable index or a variable (default: 'h').

OUTPUT:

The same algebraic form, but defined by a homogeneous polynomial.

EXAMPLES:

sage: R.<x,y,z> = QQ[]
sage: q = invariant_theory.quaternary_biquadratic(x^2+1, y^2+1, [x,y,z])
sage: q
Joint quaternary quadratic with coefficients (1, 0, 0, 1, 0, 0, 0, 0, 0, 0) 
and quaternary quadratic with coefficients (0, 1, 0, 1, 0, 0, 0, 0, 0, 0)
sage: q.homogenized()
Joint quaternary quadratic with coefficients (1, 0, 0, 1, 0, 0, 0, 0, 0, 0) 
and quaternary quadratic with coefficients (0, 1, 0, 1, 0, 0, 0, 0, 0, 0)
sage: type(q) is type(q.homogenized())
True
n_forms()

Return the number of forms.

EXAMPLES:

sage: R.<x,y> = QQ[]
sage: q1 = invariant_theory.quadratic_form(x^2 + y^2)
sage: q2 = invariant_theory.quadratic_form(x*y)
sage: from sage.rings.invariant_theory import SeveralAlgebraicForms
sage: q12 = SeveralAlgebraicForms([q1, q2])
sage: q12.n_forms()
2
sage: len(q12) == q12.n_forms()    # syntactic sugar
True
class sage.rings.invariant_theory.TernaryCubic(n, d, polynomial, *args)

Bases: sage.rings.invariant_theory.AlgebraicForm

Invariant theory of a ternary cubic.

You should use the invariant_theory factory object to contstruct instances of this class. See ternary_cubic() for details.

TESTS:

sage: R.<x,y,z> = QQ[]
sage: cubic = invariant_theory.ternary_cubic(x^3+y^3+z^3)
sage: cubic
Ternary cubic with coefficients (1, 1, 1, 0, 0, 0, 0, 0, 0, 0)
sage: TestSuite(cubic).run()
Hessian()

Return the Hessian covariant.

OUTPUT:

The Hessian matrix multiplied with the conventional normalization factor \(1/216\).

EXAMPLES:

sage: R.<x,y,z> = QQ[]
sage: cubic = invariant_theory.ternary_cubic(x^3+y^3+z^3)
sage: cubic.Hessian()
x*y*z

sage: R.<x,y> = QQ[]
sage: cubic = invariant_theory.ternary_cubic(x^3+y^3+1)
sage: cubic.Hessian()
x*y
J_covariant()

Return the J-covariant of the ternary cubic.

EXAMPLES:

sage: R.<x,y,z> = QQ[]
sage: cubic = invariant_theory.ternary_cubic(x^3+y^3+z^3)
sage: cubic.J_covariant()
x^6*y^3 - x^3*y^6 - x^6*z^3 + y^6*z^3 + x^3*z^6 - y^3*z^6

sage: R.<x,y> = QQ[]
sage: cubic = invariant_theory.ternary_cubic(x^3+y^3+1)
sage: cubic.J_covariant()
x^6*y^3 - x^3*y^6 - x^6 + y^6 + x^3 - y^3
S_invariant()

Return the S-invariant.

EXAMPLES:

sage: R.<x,y,z> = QQ[]
sage: cubic = invariant_theory.ternary_cubic(x^2*y+y^3+z^3+x*y*z)
sage: cubic.S_invariant()
-1/1296
T_invariant()

Return the T-invariant.

EXAMPLES:

sage: R.<x,y,z> = QQ[]
sage: cubic = invariant_theory.ternary_cubic(x^3+y^3+z^3)
sage: cubic.T_invariant()
1

sage: R.<x,y,z,t> = GF(7)[]
sage: cubic = invariant_theory.ternary_cubic(x^3+y^3+z^3+t*x*y*z, [x,y,z])
sage: cubic.T_invariant()
-t^6 - t^3 + 1
Theta_covariant()

Return the \(\Theta\) covariant.

EXAMPLES:

sage: R.<x,y,z> = QQ[]
sage: cubic = invariant_theory.ternary_cubic(x^3+y^3+z^3)
sage: cubic.Theta_covariant()
-x^3*y^3 - x^3*z^3 - y^3*z^3

sage: R.<x,y> = QQ[]
sage: cubic = invariant_theory.ternary_cubic(x^3+y^3+1)
sage: cubic.Theta_covariant()
-x^3*y^3 - x^3 - y^3

sage: R.<x,y,z,a30,a21,a12,a03,a20,a11,a02,a10,a01,a00> = QQ[]
sage: p = ( a30*x^3 + a21*x^2*y + a12*x*y^2 + a03*y^3 + a20*x^2*z +
...         a11*x*y*z + a02*y^2*z + a10*x*z^2 + a01*y*z^2 + a00*z^3 )
sage: cubic = invariant_theory.ternary_cubic(p, x,y,z)
sage: len(list(cubic.Theta_covariant()))
6952
coeffs()

Return the coefficients of a cubic.

Given

\[\begin{split}\begin{split} p(x,y) =&\; a_{30} x^{3} + a_{21} x^{2} y + a_{12} x y^{2} + a_{03} y^{3} + a_{20} x^{2} + \\ &\; a_{11} x y + a_{02} y^{2} + a_{10} x + a_{01} y + a_{00} \end{split}\end{split}\]

this function returns \(a = (a_{30}, a_{03}, a_{00}, a_{21}, a_{20}, a_{12}, a_{02}, a_{10}, a_{01}, a_{11})\)

EXAMPLES:

sage: R.<x,y,z,a30,a21,a12,a03,a20,a11,a02,a10,a01,a00> = QQ[]
sage: p = ( a30*x^3 + a21*x^2*y + a12*x*y^2 + a03*y^3 + a20*x^2*z +
...         a11*x*y*z + a02*y^2*z + a10*x*z^2 + a01*y*z^2 + a00*z^3 )
sage: invariant_theory.ternary_cubic(p, x,y,z).coeffs()
(a30, a03, a00, a21, a20, a12, a02, a10, a01, a11)
sage: invariant_theory.ternary_cubic(p.subs(z=1), x, y).coeffs()
(a30, a03, a00, a21, a20, a12, a02, a10, a01, a11)
monomials()

List the basis monomials of the form.

OUTPUT:

A tuple of monomials. They are in the same order as coeffs().

EXAMPLES:

sage: R.<x,y,z> = QQ[]
sage: cubic = invariant_theory.ternary_cubic(x^3+y*z^2)
sage: cubic.monomials()
(x^3, y^3, z^3, x^2*y, x^2*z, x*y^2, y^2*z, x*z^2, y*z^2, x*y*z)
polar_conic()

Return the polar conic of the cubic.

OUTPUT:

Given the ternary cubic \(f(X,Y,Z)\), this method returns the symmetric matrix \(A(x,y,z)\) defined by

\[x f_X + y f_Y + z f_Z = (X,Y,Z) \cdot A(x,y,z) \cdot (X,Y,Z)^t\]

EXAMPLES:

sage: R.<x,y,z,X,Y,Z,a30,a21,a12,a03,a20,a11,a02,a10,a01,a00> = QQ[]
sage: p = ( a30*x^3 + a21*x^2*y + a12*x*y^2 + a03*y^3 + a20*x^2*z +
...         a11*x*y*z + a02*y^2*z + a10*x*z^2 + a01*y*z^2 + a00*z^3 )
sage: cubic = invariant_theory.ternary_cubic(p, x,y,z)
sage: cubic.polar_conic()
[  3*x*a30 + y*a21 + z*a20 x*a21 + y*a12 + 1/2*z*a11 x*a20 + 1/2*y*a11 + z*a10]
[x*a21 + y*a12 + 1/2*z*a11   x*a12 + 3*y*a03 + z*a02 1/2*x*a11 + y*a02 + z*a01]
[x*a20 + 1/2*y*a11 + z*a10 1/2*x*a11 + y*a02 + z*a01   x*a10 + y*a01 + 3*z*a00]

sage: polar_eqn = X*p.derivative(x) + Y*p.derivative(y) + Z*p.derivative(z)
sage: polar = invariant_theory.ternary_quadratic(polar_eqn, [x,y,z])
sage: polar.matrix().subs(X=x,Y=y,Z=z) == cubic.polar_conic()
True
scaled_coeffs()

Return the coefficients of a cubic.

Compared to coeffs(), this method returns rescaled coefficients that are often used in invariant theory.

Given

\[\begin{split}\begin{split} p(x,y) =&\; a_{30} x^{3} + a_{21} x^{2} y + a_{12} x y^{2} + a_{03} y^{3} + a_{20} x^{2} + \\ &\; a_{11} x y + a_{02} y^{2} + a_{10} x + a_{01} y + a_{00} \end{split}\end{split}\]

this function returns \(a = (a_{30}, a_{03}, a_{00}, a_{21}/3, a_{20}/3, a_{12}/3, a_{02}/3, a_{10}/3, a_{01}/3, a_{11}/6)\)

EXAMPLES:

sage: R.<x,y,z,a30,a21,a12,a03,a20,a11,a02,a10,a01,a00> = QQ[]
sage: p = ( a30*x^3 + a21*x^2*y + a12*x*y^2 + a03*y^3 + a20*x^2*z +
...         a11*x*y*z + a02*y^2*z + a10*x*z^2 + a01*y*z^2 + a00*z^3 )
sage: invariant_theory.ternary_cubic(p, x,y,z).scaled_coeffs()
(a30, a03, a00, 1/3*a21, 1/3*a20, 1/3*a12, 1/3*a02, 1/3*a10, 1/3*a01, 1/6*a11)
syzygy(U, S, T, H, Theta, J)

Return the syzygy of the cubic evaluated on the invariants and covariants.

INPUT:

  • U, S, T, H, Theta, J – polynomials from the same polynomial ring.

OUTPUT:

0 if evaluated for the form, the S invariant, the T invariant, the Hessian, the \(\Theta\) covariant and the J-covariant of a ternary cubic.

EXAMPLES:

sage: R.<x,y,z> = QQ[]
sage: monomials = (x^3, y^3, z^3, x^2*y, x^2*z, x*y^2,
...                y^2*z, x*z^2, y*z^2, x*y*z)
sage: random_poly = sum([ randint(0,10000) * m for m in monomials ])
sage: cubic = invariant_theory.ternary_cubic(random_poly)
sage: U = cubic.form()
sage: S = cubic.S_invariant()
sage: T = cubic.T_invariant()
sage: H = cubic.Hessian()
sage: Theta = cubic.Theta_covariant()
sage: J = cubic.J_covariant() 
sage: cubic.syzygy(U, S, T, H, Theta, J)
0
class sage.rings.invariant_theory.TernaryQuadratic(n, d, polynomial, *args)

Bases: sage.rings.invariant_theory.QuadraticForm

Invariant theory of a ternary quadratic.

You should use the invariant_theory factory object to construct instances of this class. See ternary_quadratic() for details.

TESTS:

sage: R.<x,y,z> = QQ[]
sage: quadratic = invariant_theory.ternary_quadratic(x^2+y^2+z^2)
sage: quadratic
Ternary quadratic with coefficients (1, 1, 1, 0, 0, 0)
sage: TestSuite(quadratic).run()
coeffs()

Return the coefficients of a quadratic.

Given

\[\begin{split}p(x,y) =&\; a_{20} x^{2} + a_{11} x y + a_{02} y^{2} + a_{10} x + a_{01} y + a_{00}\end{split}\]

this function returns \(a = (a_{20}, a_{02}, a_{00}, a_{11}, a_{10}, a_{01} )\)

EXAMPLES:

sage: R.<x,y,z,a20,a11,a02,a10,a01,a00> = QQ[]
sage: p = ( a20*x^2 + a11*x*y + a02*y^2 +
...         a10*x*z + a01*y*z + a00*z^2 )
sage: invariant_theory.ternary_quadratic(p, x,y,z).coeffs()
(a20, a02, a00, a11, a10, a01)
sage: invariant_theory.ternary_quadratic(p.subs(z=1), x, y).coeffs()
(a20, a02, a00, a11, a10, a01)
covariant_conic(other)

Return the ternary quadratic covariant to self and other.

INPUT:

  • other – Another ternary quadratic.

OUTPUT:

The so-called covariant conic, a ternary quadratic. It is symmetric under exchange of self and other.

EXAMPLES:

sage: ring.<x,y,z> = QQ[]
sage: Q = invariant_theory.ternary_quadratic(x^2+y^2+z^2)
sage: R = invariant_theory.ternary_quadratic(x*y+x*z+y*z)
sage: Q.covariant_conic(R)
-x*y - x*z - y*z
sage: R.covariant_conic(Q)
-x*y - x*z - y*z

TESTS:

sage: R.<a,a_,b,b_,c,c_,f,f_,g,g_,h,h_,x,y,z> = QQ[]
sage: p = ( a*x^2 + 2*h*x*y + b*y^2 +
...         2*g*x*z + 2*f*y*z + c*z^2 )
sage: Q = invariant_theory.ternary_quadratic(p, [x,y,z])
sage: Q.matrix()
[a h g]
[h b f]
[g f c]
sage: p = ( a_*x^2 + 2*h_*x*y + b_*y^2 +
...         2*g_*x*z + 2*f_*y*z + c_*z^2 )
sage: Q_ = invariant_theory.ternary_quadratic(p, [x,y,z])
sage: Q_.matrix()
[a_ h_ g_]
[h_ b_ f_]
[g_ f_ c_]
sage: QQ_ = Q.covariant_conic(Q_)
sage: invariant_theory.ternary_quadratic(QQ_, [x,y,z]).matrix()
[      b_*c + b*c_ - 2*f*f_  f_*g + f*g_ - c_*h - c*h_ -b_*g - b*g_ + f_*h + f*h_]
[ f_*g + f*g_ - c_*h - c*h_       a_*c + a*c_ - 2*g*g_ -a_*f - a*f_ + g_*h + g*h_]
[-b_*g - b*g_ + f_*h + f*h_ -a_*f - a*f_ + g_*h + g*h_       a_*b + a*b_ - 2*h*h_]
monomials()

List the basis monomials of the form.

OUTPUT:

A tuple of monomials. They are in the same order as coeffs().

EXAMPLES:

sage: R.<x,y,z> = QQ[]
sage: quadratic = invariant_theory.ternary_quadratic(x^2+y*z)
sage: quadratic.monomials()
(x^2, y^2, z^2, x*y, x*z, y*z)
scaled_coeffs()

Return the scaled coefficients of a quadratic.

Given

\[\begin{split}p(x,y) =&\; a_{20} x^{2} + a_{11} x y + a_{02} y^{2} + a_{10} x + a_{01} y + a_{00}\end{split}\]

this function returns \(a = (a_{20}, a_{02}, a_{00}, a_{11}/2, a_{10}/2, a_{01}/2, )\)

EXAMPLES:

sage: R.<x,y,z,a20,a11,a02,a10,a01,a00> = QQ[]
sage: p = ( a20*x^2 + a11*x*y + a02*y^2 +
...         a10*x*z + a01*y*z + a00*z^2 )
sage: invariant_theory.ternary_quadratic(p, x,y,z).scaled_coeffs()
(a20, a02, a00, 1/2*a11, 1/2*a10, 1/2*a01)
sage: invariant_theory.ternary_quadratic(p.subs(z=1), x, y).scaled_coeffs()
(a20, a02, a00, 1/2*a11, 1/2*a10, 1/2*a01)
class sage.rings.invariant_theory.TwoAlgebraicForms(forms)

Bases: sage.rings.invariant_theory.SeveralAlgebraicForms

The Python constructor.

TESTS:

sage: from sage.rings.invariant_theory import AlgebraicForm, SeveralAlgebraicForms
sage: R.<x,y,z> = QQ[]
sage: p = AlgebraicForm(2, 2, x^2 + y^2)
sage: q = AlgebraicForm(2, 3, x^3 + y^3)
sage: r = AlgebraicForm(3, 3, x^3 + y^3 + z^3)
sage: pq = SeveralAlgebraicForms([p, q])
sage: pr = SeveralAlgebraicForms([p, r])
Traceback (most recent call last):
...
ValueError: All forms must be in the same variables.
first()

Return the first of the two forms.

OUTPUT:

The first algebraic form used in the definition.

EXAMPLES:

sage: R.<x,y> = QQ[]
sage: q0 = invariant_theory.quadratic_form(x^2 + y^2)
sage: q1 = invariant_theory.quadratic_form(x*y)
sage: from sage.rings.invariant_theory import TwoAlgebraicForms
sage: q = TwoAlgebraicForms([q0, q1])
sage: q.first() is q0
True
sage: q.get_form(0) is q0
True
sage: q.first().polynomial()
x^2 + y^2
second()

Return the second of the two forms.

OUTPUT:

The second form used in the definition.

EXAMPLES:

sage: R.<x,y> = QQ[]
sage: q0 = invariant_theory.quadratic_form(x^2 + y^2)
sage: q1 = invariant_theory.quadratic_form(x*y)
sage: from sage.rings.invariant_theory import TwoAlgebraicForms
sage: q = TwoAlgebraicForms([q0, q1])
sage: q.second() is q1
True
sage: q.get_form(1) is q1
True
sage: q.second().polynomial()
x*y
class sage.rings.invariant_theory.TwoQuaternaryQuadratics(forms)

Bases: sage.rings.invariant_theory.TwoAlgebraicForms

Invariant theory of two quaternary quadratics.

You should use the invariant_theory factory object to construct instances of this class. See quaternary_biquadratics() for details.

REFERENCES:

[Salmon]G. Salmon: “A Treatise on the Analytic Geometry of Three Dimensions”, section on “Invariants and Covariants of Systems of Quadrics”.

TESTS:

sage: R.<w,x,y,z> = QQ[]
sage: inv = invariant_theory.quaternary_biquadratic(w^2+x^2, y^2+z^2, w, x, y, z)
sage: inv
Joint quaternary quadratic with coefficients (1, 1, 0, 0, 0, 0, 0, 0, 0, 0) and 
quaternary quadratic with coefficients (0, 0, 1, 1, 0, 0, 0, 0, 0, 0)
sage: TestSuite(inv).run()

sage: q1 = 73*x^2 + 96*x*y - 11*y^2 - 74*x*z - 10*y*z + 66*z^2 + 4*x + 63*y - 11*z + 57
sage: q2 = 61*x^2 - 100*x*y - 72*y^2 - 38*x*z + 85*y*z + 95*z^2 - 81*x + 39*y + 23*z - 7
sage: biquadratic = invariant_theory.quaternary_biquadratic(q1, q2, [x,y,z]).homogenized()
sage: biquadratic._check_covariant('Delta_invariant', invariant=True)
sage: biquadratic._check_covariant('Delta_prime_invariant', invariant=True)
sage: biquadratic._check_covariant('Theta_invariant', invariant=True)
sage: biquadratic._check_covariant('Theta_prime_invariant', invariant=True)
sage: biquadratic._check_covariant('Phi_invariant', invariant=True)
sage: biquadratic._check_covariant('T_covariant')
sage: biquadratic._check_covariant('T_prime_covariant')
sage: biquadratic._check_covariant('J_covariant')
Delta_invariant()

Return the \(\Delta\) invariant.

EXAMPLES:

sage: R.<x,y,z,t,a0,a1,a2,a3,b0,b1,b2,b3,b4,b5,A0,A1,A2,A3,B0,B1,B2,B3,B4,B5> = QQ[]
sage: p1 = a0*x^2 + a1*y^2 + a2*z^2 + a3
sage: p1 += b0*x*y + b1*x*z + b2*x + b3*y*z + b4*y + b5*z
sage: p2 = A0*x^2 + A1*y^2 + A2*z^2 + A3
sage: p2 += B0*x*y + B1*x*z + B2*x + B3*y*z + B4*y + B5*z
sage: q = invariant_theory.quaternary_biquadratic(p1, p2, [x, y, z])
sage: coeffs = det(t * q[0].matrix() + q[1].matrix()).polynomial(t).coeffs()
sage: q.Delta_invariant() == coeffs[4]
True
Delta_prime_invariant()

Return the \(\Delta'\) invariant.

EXAMPLES:

sage: R.<x,y,z,t,a0,a1,a2,a3,b0,b1,b2,b3,b4,b5,A0,A1,A2,A3,B0,B1,B2,B3,B4,B5> = QQ[]
sage: p1 = a0*x^2 + a1*y^2 + a2*z^2 + a3
sage: p1 += b0*x*y + b1*x*z + b2*x + b3*y*z + b4*y + b5*z
sage: p2 = A0*x^2 + A1*y^2 + A2*z^2 + A3
sage: p2 += B0*x*y + B1*x*z + B2*x + B3*y*z + B4*y + B5*z
sage: q = invariant_theory.quaternary_biquadratic(p1, p2, [x, y, z])
sage: coeffs = det(t * q[0].matrix() + q[1].matrix()).polynomial(t).coeffs()
sage: q.Delta_prime_invariant() == coeffs[0]
True
J_covariant()

The \(J\)-covariant.

This is the Jacobian determinant of the two biquadratics, the \(T\)-covariant, and the \(T'\)-covariant with respect to the four homogeneous variables.

EXAMPLES:

sage: R.<w,x,y,z,a0,a1,a2,a3,A0,A1,A2,A3> = QQ[]
sage: p1 = a0*x^2 + a1*y^2 + a2*z^2 + a3*w^2
sage: p2 = A0*x^2 + A1*y^2 + A2*z^2 + A3*w^2
sage: q = invariant_theory.quaternary_biquadratic(p1, p2, [w, x, y, z])
sage: q.J_covariant().factor()
z * y * x * w * (a3*A2 - a2*A3) * (a3*A1 - a1*A3) * (-a2*A1 + a1*A2)
* (a3*A0 - a0*A3) * (-a2*A0 + a0*A2) * (-a1*A0 + a0*A1)
Phi_invariant()

Return the \(\Phi'\) invariant.

EXAMPLES:

sage: R.<x,y,z,t,a0,a1,a2,a3,b0,b1,b2,b3,b4,b5,A0,A1,A2,A3,B0,B1,B2,B3,B4,B5> = QQ[]
sage: p1 = a0*x^2 + a1*y^2 + a2*z^2 + a3
sage: p1 += b0*x*y + b1*x*z + b2*x + b3*y*z + b4*y + b5*z
sage: p2 = A0*x^2 + A1*y^2 + A2*z^2 + A3
sage: p2 += B0*x*y + B1*x*z + B2*x + B3*y*z + B4*y + B5*z
sage: q = invariant_theory.quaternary_biquadratic(p1, p2, [x, y, z])
sage: coeffs = det(t * q[0].matrix() + q[1].matrix()).polynomial(t).coeffs()
sage: q.Phi_invariant() == coeffs[2]
True
T_covariant()

The \(T\)-covariant.

EXAMPLES:

sage: R.<x,y,z,t,a0,a1,a2,a3,b0,b1,b2,b3,b4,b5,A0,A1,A2,A3,B0,B1,B2,B3,B4,B5> = QQ[]
sage: p1 = a0*x^2 + a1*y^2 + a2*z^2 + a3
sage: p1 += b0*x*y + b1*x*z + b2*x + b3*y*z + b4*y + b5*z
sage: p2 = A0*x^2 + A1*y^2 + A2*z^2 + A3
sage: p2 += B0*x*y + B1*x*z + B2*x + B3*y*z + B4*y + B5*z
sage: q = invariant_theory.quaternary_biquadratic(p1, p2, [x, y, z])
sage: T = invariant_theory.quaternary_quadratic(q.T_covariant(), [x,y,z]).matrix()
sage: M = q[0].matrix().adjoint() + t*q[1].matrix().adjoint()
sage: M = M.adjoint().apply_map(             # long time (4s on my thinkpad W530)
....:         lambda m: m.coefficient(t))
sage: M == q.Delta_invariant()*T             # long time
True
T_prime_covariant()

The \(T'\)-covariant.

EXAMPLES:

sage: R.<x,y,z,t,a0,a1,a2,a3,b0,b1,b2,b3,b4,b5,A0,A1,A2,A3,B0,B1,B2,B3,B4,B5> = QQ[]
sage: p1 = a0*x^2 + a1*y^2 + a2*z^2 + a3
sage: p1 += b0*x*y + b1*x*z + b2*x + b3*y*z + b4*y + b5*z
sage: p2 = A0*x^2 + A1*y^2 + A2*z^2 + A3
sage: p2 += B0*x*y + B1*x*z + B2*x + B3*y*z + B4*y + B5*z
sage: q = invariant_theory.quaternary_biquadratic(p1, p2, [x, y, z])
sage: Tprime = invariant_theory.quaternary_quadratic(
....:     q.T_prime_covariant(), [x,y,z]).matrix()
sage: M = q[0].matrix().adjoint() + t*q[1].matrix().adjoint()
sage: M = M.adjoint().apply_map(                # long time (4s on my thinkpad W530)
....:         lambda m: m.coefficient(t^2))
sage: M == q.Delta_prime_invariant() * Tprime   # long time
True
Theta_invariant()

Return the \(\Theta\) invariant.

EXAMPLES:

sage: R.<x,y,z,t,a0,a1,a2,a3,b0,b1,b2,b3,b4,b5,A0,A1,A2,A3,B0,B1,B2,B3,B4,B5> = QQ[]
sage: p1 = a0*x^2 + a1*y^2 + a2*z^2 + a3
sage: p1 += b0*x*y + b1*x*z + b2*x + b3*y*z + b4*y + b5*z
sage: p2 = A0*x^2 + A1*y^2 + A2*z^2 + A3
sage: p2 += B0*x*y + B1*x*z + B2*x + B3*y*z + B4*y + B5*z
sage: q = invariant_theory.quaternary_biquadratic(p1, p2, [x, y, z])
sage: coeffs = det(t * q[0].matrix() + q[1].matrix()).polynomial(t).coeffs()
sage: q.Theta_invariant() == coeffs[3]
True
Theta_prime_invariant()

Return the \(\Theta'\) invariant.

EXAMPLES:

sage: R.<x,y,z,t,a0,a1,a2,a3,b0,b1,b2,b3,b4,b5,A0,A1,A2,A3,B0,B1,B2,B3,B4,B5> = QQ[]
sage: p1 = a0*x^2 + a1*y^2 + a2*z^2 + a3
sage: p1 += b0*x*y + b1*x*z + b2*x + b3*y*z + b4*y + b5*z
sage: p2 = A0*x^2 + A1*y^2 + A2*z^2 + A3
sage: p2 += B0*x*y + B1*x*z + B2*x + B3*y*z + B4*y + B5*z
sage: q = invariant_theory.quaternary_biquadratic(p1, p2, [x, y, z])
sage: coeffs = det(t * q[0].matrix() + q[1].matrix()).polynomial(t).coeffs()
sage: q.Theta_prime_invariant() == coeffs[1]
True
syzygy(Delta, Theta, Phi, Theta_prime, Delta_prime, U, V, T, T_prime, J)

Return the syzygy evaluated on the invariants and covariants.

INPUT:

  • Delta, Theta, Phi, Theta_prime, Delta_prime, U, V, T, T_prime, J – polynomials from the same polynomial ring.

OUTPUT:

Zero if the U is the first polynomial, V the second polynomial, and the remaining input are the invariants and covariants of a quaternary biquadratic.

EXAMPLES:

sage: R.<w,x,y,z> = QQ[]
sage: monomials = [x^2, x*y, y^2, x*z, y*z, z^2, x*w, y*w, z*w, w^2]
sage: def q_rnd():  return sum(randint(-1000,1000)*m for m in monomials)
sage: biquadratic = invariant_theory.quaternary_biquadratic(q_rnd(), q_rnd())
sage: Delta = biquadratic.Delta_invariant()
sage: Theta = biquadratic.Theta_invariant()
sage: Phi = biquadratic.Phi_invariant()
sage: Theta_prime = biquadratic.Theta_prime_invariant()
sage: Delta_prime = biquadratic.Delta_prime_invariant()
sage: U = biquadratic.first().polynomial()
sage: V  = biquadratic.second().polynomial()
sage: T = biquadratic.T_covariant()
sage: T_prime  = biquadratic.T_prime_covariant()
sage: J = biquadratic.J_covariant()
sage: biquadratic.syzygy(Delta, Theta, Phi, Theta_prime, Delta_prime, U, V, T, T_prime, J)
0

If the arguments are not the invariants and covariants then the output is some (generically non-zero) polynomial:

sage: biquadratic.syzygy(1, 1, 1, 1, 1, 1, 1, 1, 1, x)
-x^2 + 1

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