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# Classical Invariant Theory Through an Example

## Classical Invariant Theory

Classical invariant theory is the study of intrinsic properties of polynomials.
By intrinsic, we mean those properties which are unaffected by a change of
variables and are purely geometric.

Intrinsic properties:
• factorizability,
• multiplicities of roots,
• geometrical configurations of roots.
Non-intrinsic properties:
• explicit values of the roots,
• particular coefficients of the polynomial

The study of invariants is closely tied to

the problem of equivalence:
• when can one polynomial be transformed into another by a suitable
change of coordinates.

the associated canonical form problem:
• find a system of coordinates in which the polynomial takes on a
particular simple form.

The first goal of classical invariant theory is to determine the fundamental invariants of polynomials.

## George Boole (November 2, 1815 { December 8, 1864) was an English mathematician and philosopher.  Inventor of Boolean algebra

## Arthur Cayley (August 16, 1821 { January 26, 1895) was a British mathematician.  Proved the Cayley-Hamilton theorem: every square matrix is a root of its own characteristic polynomial.   He was the first to define the concept of a group in the modern way: as a set with a binary operation satisfying certain laws.

## David Hilbert (January 23, 1862 { February 14, 1943) was a German mathematician.  Proved the Hilbert basis theorem.   Invented Hibert spaces. The 23 Problems

Disclaimer: All the considerations are over C

Let be a quadratic polynomial. As long as a ≠ 0, the roots are where The existence of two roots implies that we can write Proposition The affine transformation preserves the class of quadratic polynomials

An affine transformation sends a polynomial p(x) to a new polynomial defined so that Say then Expending we get Hence and Example Consider the polynomial and the change of variable Then The roots of p go to and ## Canonical Forms

If then p has 2 distinct roots. By translation we can shift x_
to 0, then by (complex) dilation we can make x+ to be equal to 2i .
Then we can shift the complex plane by -i . Hence a canonical
polynomial is If then p has a double root x0. We can translate x0 to 0, thus is a multiple of . By scaling we can reduce the multiple to 1.
Hence the canonical form is Proposition The affine canonical forms for complex quadratic polynomials are Definition The homogeneous quadratic polynomial in two variables x, y is called a quadratic form.

To every quadratic form we can associate the polynomial Inversely, a polynomial p(z) induces a quadratic form Theorem Any invertible change of variables of the form maps a quadratic form to a quadratic form according to Let The coefficients are related to the coefficients a, b, and c of
q(x, y) by the relations Definition An invariant of a quadratic form q(x, y) is a function I (a, b, c), which, up to a determinantal factor, does not change under a general linear transformation: The determinantal power k is called the weight of the invariant Theorem The discriminant is an invariant of weight two: Remark Our classification of quadratic polynomials is based on the fact that = 0 or . From (1), if then it stays equal (not equal) to zero under a change of variables

## Projective Transformations

Recall that Thus the transformation induces the transformation where z = x/y.

 Definition The transformation is called a linear fractional or MÖbius or projective transformation.

This simple transformation is of fundamental importance in

•projective geometry,
•complex analysis and geometry,
•number theory,
•hyperbolic geometry.

A linear fractional transformation induces a transformation on quadratic
polynomials defined by The transformation rules for the coefficients of a quadratic polynomial are Example The inversion maps the quadratic polynomial to Thus projective transformations do not necessarily preserve the degree of a polynomial.

## Canonical forms

Let p(z) be a quadratic polynomial. We can assume it is in one of the affine canonical forms.

1) Let p(z) = k(z^2 + 1). If we scale according to the coefficient matrix such that λ^2 = k then Under the transformation the polynomial z^2 + 1 goes to 2) Let p(z) = z^2 then under the inversion it is mapped to Proposition The canonical forms for complex quadratic polynomials are ## Generalizations

•Polynomials and forms of order greater than 2.
•Polynomials and forms with more variables (hardcore algebraic geometry!).
•Work with polynomial rings over more general fields (hardcore algebra!).

## (Partially) Open Problems

1) The discriminant  is an invariant and characterizes the multiplicity of the roots (and the
canonical forms).

Question: If one considers polynomials of higher degree or larger number
of variables, how many invariants, similar to the discriminant, are there?
What properties do they characterize?

2) Consider the binary form Any of the following four complex linear substitution does not change the polynomial. p(x, y) is also invariant under the group
they generate, consisting of 192 elements.

The binary form has infinitely many symmetries since it contains all rotations in the
xy-plane.

The binary form is preserved only by scaling by a fifth root of unity: Question: Given a multivariable polynomial, how can one efficiently find
the size of its symmetry group and compute it explicitly?

3) The transformation sends to Note that the Hessian Fact: The Hessian of a homogeneous polynomial in 2 variables is zero if
and only if it can be transformed to a polynomial of a single variable by a
linear change of variables.

Hesse claimed a similar result was true for any number of variables: A
homogeneous polynomial can be transformed to a polynomial of fewer than m variables if and only if
its Hessian The conjecture was shown to be true by Noether and Gordan only for m≤4.

Question: How can one determine efficiently that a given polynomial
essentially depends on a fewer number of variables than it seems to be?
How to find the corresponding change of variables.