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.
Nonintrinsic 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.
(November 2, 1815 { December 8, 1864) was an English mathematician and philosopher. Inventor of Boolean algebra 
(August 16, 1821 { January 26, 1895) was a British mathematician. Proved the CayleyHamilton He was the first to define the 
(January 23, 1862 { February 14, 1943) was a German mathematician. Proved the Hilbert basis Invented Hibert spaces. 
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 
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 x_{0}. We can translate x_{0} 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 
Recall that
Thus the transformation
induces the transformation
where z = x/y.
Definition 
The transformation
is called a linear fractional or MÃ–bius 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. 
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

•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!).
1) The discriminant
of a quadratic polynomial
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
xyplane.
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.