Hessian and inflexion points
Theorem 1.
Suppose that is a curve in the real projective plane
given by a homogeneous
equation of
degree of homogeneity (http://planetmath.org/HomogeneousFunction) .
If has continuous first derivatives
in a neighborhood of
a point and the gradient of is non-zero at and
is an inflection point
of , then , where is
the Hessian determinant:
Proof.
We may choose a system of homogenous coordinates
such that the point lies at and the equation
of the tangent to at is . Using the
implicit function theorem, we may conclude that there
exists an interval
and a function
such that
when . In
other words, the portion of curve near may be described
in non-homogenous coordinates by . By the way
the coordinates were chosen, and .
Because is an inflection point, we also have .
Differentiating the equation twice, we obtain the following:
We will now put but, for reasons which will be explained later, we do not yet want to make use of the fact that :
Since is homogenous, Euler’s formula holds:
Taking partial derivatives, we obtain the following:
Evaluating at and making use of the equations deduced above, we obtain the following:
Making use of these facts, we may now evaluate the determinant:
Since is an inflection point, , so we have . ∎
Actually, we proved slightly more than what was stated.
Because the gradient is assumed not to vanish at ,
but and by the way we set up our coordinate
system, we must have .
Thus, we see that, if , then
if and only if . However, note that this does
not mean that the Hessian vanishes if and only if is
an inflection point since the definition of inflection
point not only requires that but that the
sign of change as passes through .
This result is used quite often in algebraic geometry,
where is a homogenous polynomial
. In such a context,
it is desirable to keep demonstrations purely
algebraic and avoid introducing analysis
where possible,
so a variant of this result is preferred. The theorem
may be restated as follows:
Theorem 2.
Suppose that is a curve in the real projective plane
given by an equation
where is a homogenous polynomial of degree .
If is regular at a point and is an inflection point
of , then , where is the Hessian determinant.
To make our proof purely algebraic, we replace the use of
the implicit function theorem to obtain with an expansion
in a formal power series. As above, we choose our
coordinates so as to place at and make
tangent to the line at . Then, since is a regular
point of , we may parameterize by a formal power series
such that .
Then, if we
define derivatives algebraically (http://planetmath.org/DerivativeOfPolynomial),
we may proceed with the rest of the proof exactly as above.
Title | Hessian and inflexion points |
---|---|
Canonical name | HessianAndInflexionPoints |
Date of creation | 2013-03-22 18:22:26 |
Last modified on | 2013-03-22 18:22:26 |
Owner | rspuzio (6075) |
Last modified by | rspuzio (6075) |
Numerical id | 14 |
Author | rspuzio (6075) |
Entry type | Theorem |
Classification | msc 53A04 |
Classification | msc 26A51 |