# How Do Inflection Points Intersect with the Hessian on Algebraic Curves?

• MHB
• evinda
In summary, to find the non-singular points of an algebraic curve that intersect with the hessian, we can use partial derivatives and the hessian matrix to determine the critical points. In the case of the given curve $x^3+y^3+z^3=0$, there are no non-singular points that intersect with the hessian.
evinda
Gold Member
MHB
Hi! (Smile)

Let the algebraic curve $f(x_0, x_1, x_2) \in K[x_0, x_1, x_2]$. The inflection points are the non-singular points of the curve that are the intersection points with the hessian.

If we have the curve $x^3+y^3+z^3=0$ the hessian is equal to $216 \cdot x \cdot y \cdot z$. How can we find the non-singular points of the curve that are the intersection points with the hessian? (Thinking)

Hello! It's great to see your interest in algebraic curves and their inflection points. To answer your question, we can use the concept of partial derivatives to find the non-singular points of the curve that intersect with the hessian.

First, let's rewrite the given curve as $f(x_0, x_1, x_2) = x_0^3 + x_1^3 + x_2^3 = 0$. Now, we can take partial derivatives of this function with respect to each variable, giving us:

$\frac{\partial f}{\partial x_0} = 3x_0^2$
$\frac{\partial f}{\partial x_1} = 3x_1^2$
$\frac{\partial f}{\partial x_2} = 3x_2^2$

Next, we can set these partial derivatives equal to 0 and solve for $x_0, x_1, x_2$ to find the critical points of the curve. In this case, we get $x_0 = x_1 = x_2 = 0$.

Now, we can use the hessian matrix, which is the matrix of second partial derivatives of our function, to determine the type of critical point at $(x_0, x_1, x_2) = (0,0,0)$. The hessian matrix for our curve is:

$H = \begin{bmatrix} 6x_0 & 0 & 0 \\ 0 & 6x_1 & 0 \\ 0 & 0 & 6x_2 \end{bmatrix} = \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}$

This matrix has a determinant of 0, which means that it is a singular point on the curve. In other words, the curve does not have any non-singular points that intersect with the hessian.

## What does it mean to find non-singular points?

Finding non-singular points refers to identifying and locating points on a mathematical curve or surface where the derivative or gradient is not equal to zero. These points are also known as critical points or stationary points.

## Why is it important to find non-singular points?

Non-singular points play a crucial role in mathematical analysis and optimization problems. They can help determine the maximum or minimum values of a function, as well as the direction of its change. In some cases, these points can also reveal important information about the behavior of a system or process.

## How do you find non-singular points?

The process of finding non-singular points varies depending on the type of curve or surface being analyzed. In general, it involves taking the derivative or gradient of the function, setting it equal to zero, and solving for the variables. This will result in a set of equations that can be solved to determine the coordinates of the non-singular points.

## What are some real-world applications of finding non-singular points?

Non-singular points have numerous applications in various fields, including economics, physics, engineering, and computer science. They can be used to optimize processes and systems, such as finding the most efficient route for a delivery truck, or determining the optimal placement of wind turbines to generate maximum energy.

## Can non-singular points exist in higher dimensions?

Yes, non-singular points can exist in any number of dimensions. In fact, the concept of non-singular points extends beyond just two or three dimensions and is used in higher-level mathematical theories such as calculus of variations and differential geometry.

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