Hessian and inflexion points
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:
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 :
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:
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|
|Date of creation||2013-03-22 18:22:26|
|Last modified on||2013-03-22 18:22:26|
|Last modified by||rspuzio (6075)|