Question

A simple point \(P\) on a curve \(F\) is called a flex if \(\operatorname{ord}_{P}^{F}(L) \geq 3\), where \(L\) is the tangent to \(F\) at \(P\). The flex is called ordinary if \(\operatorname{ord}_{P}(L)=3\), a higher flex otherwise. (a) Let \(F=Y-X^{n} .\) For which \(n\) does \(F\) have a flex at \(P=(0,0)\), and what kind of flex? (b) Suppose \(P=(0,0), L=Y\) is the tangent line, \(F=Y+a X^{2}+\cdots .\) Show that \(P\) is a flex on \(F\) if and only if \(a=0 .\) Give a simple criterion for calculating ord \(_{P}^{F}(Y)\), and therefore for determining if \(P\) is a higher flex.

Short answer

Question: Determine the value of n for which the curve F has a flex at P=(0,0) and the type of flex, given that F = Y - X^n. Show that P is a flex on F if and only if a=0, given that F = Y + aX^2 + ... . Lastly, find a criterion for calculating ord\(_{P}^{F}(Y)\). Answer: The curve F has a flex at P=(0,0) for n=3 (ordinary flex) and for n>3 (higher flex). The point P is a flex on F if and only if a=0. The criterion for calculating ord\(_{P}^{F}(Y)\) is that the order of contact of the tangent line and the curve at P equals the power of the lowest-degree non-zero term in the Taylor Series expansion after the Y term.

Step-by-step solution

Part (a) - Derive the first and second derivatives of \(F\)

In order to find the order of the tangent line, it's necessary to calculate the derivatives of the curve \(F\). Let's find the first and second derivatives of the curve \(F = Y - X^n\) with respect to \(X\). First derivative: \(F'(X) = -nX^{n-1}\). Second derivative: \(F''(X) = -n(n-1)X^{n-2}\).

Part (a) - Evaluate the derivatives at the point P

Now let's evaluate the first and second derivatives of \(F\) at the point \(P = (0,0)\). First derivative at \(P(0,0)\): \(F'(0) = -n(0)^{n-1} = 0\). Second derivative at \(P(0,0)\): \(F''(0) = -n(n-1)(0)^{n-2} = 0\). Since both the first and second derivatives are \(0\) at the point \(P\), it implies that \(P\) lies on the curve \(F=Y-X^n\), as the tangent will coincide with the curve's direction.

Part (a) - Determine the value of n and type of flex

To determine the value of n for which \(F\) has a flex at \(P=(0,0)\), we need to check if \(\operatorname{ord}_{P}^{F}(L) \geq 3\). In this case, since both the first and second derivatives are identically \(0\), we have an ordinary flex if \(n=3\) and higher flex if \(n>3\). Therefore, \(F\) has a flex at \(P=(0,0)\) for \(n=3\) (ordinary) and \(n>3\) (higher).

Part (b) - Show that P is a flex on F if and only if a = 0

To prove that \(P\) is a flex on \(F\) if and only if \(a=0\), we need to find the order of contact of the tangent line \(L=Y\) to the curve \(F=Y+aX^2+\cdots\) at \(P=(0,0)\). Comparing coefficients of \(X\) and \(X^2\), we have: \(Y = Y + aX^2 + \cdots\) Setting \(X=0\), we get \(Y=Y\). This tells us that the order of contact of \(L\) and \(F\) at \(P\) is at least 1. Now, in order for \(P\) to be a flex on \(F\), we need the order of contact to be at least 3. Comparing \(X^2\) terms, we get \(aX^2 = aX^2\). Thus, the order of contact of \(L\) and \(F\) at \(P\) increases only when \(a=0\). Therefore, \(P\) is a flex on \(F\) if and only if \(a=0\).

Part (b) - Criterion for calculating ord\(_{P}^{F}(Y)\)

To find a simple criterion for calculating \(\operatorname{ord}_{P}^{F}(Y)\), we need to analyze the terms in the Taylor Series expansion of \(F(Y)\). The Taylor Series expansion around \(P=(0,0)\) is: \(F(Y) = Y + aX^2 + \cdots\) The first non-zero term in the expansion is \(Y\). So, the order of contact of \(L\) and \(F\) at \(P\) is equal to the power of the lowest-degree non-zero term in the expansion after the \(Y\) term. If this power is 3 or more, then \(P\) is a higher flex; otherwise, \(P\) is an ordinary flex.

Key concepts

Simple Point

In the realm of algebraic geometry, when we talk about a "simple point" on a curve, we're referring to a point where the curve is smooth and has no self-intersections. This point behaves like a normal, smooth part of the curve — kind of like a gentle bend or arc. At a simple point, the curve doesn’t cross itself, and it isn’t twisted or pinched, making it easier to analyze geometrically.

The mathematical properties of a simple point include:

• The tangent line exists at that point.
• The curve is differentiable, which means we can take derivatives as we would for a standard function.

At a simple point, the tangent line touches the curve lightly without any abrupt changes in direction.

Flex Point

A flex point, also known as a flex or inflection point, is more special than just any simple point. It’s where the curve changes its curvature or bends in a different direction. Mathematically, at a flex point, the tangent line has a contact order of at least 3 with the curve — this means the curve flattens out more at such points.

For a curve expressed as a polynomial, determining a flex involves:

• Finding the derivatives of the function.
• Ensuring that higher-order derivatives satisfy specific conditions; for instance, in our scenario, when a curve like \(F=Y-X^n\) at point \(P=(0,0)\), flex points are determined based on the behavior of derivatives higher than the first and second being identically zero, which indicates an ordinary flex if the condition holds for \(n=3\) and a higher flex if \(n>3\).

Flex points are notable because they indicate where the curve shifts from being concave to convex, or vice versa.

Tangent Line

The tangent line is a core concept in both calculus and geometry. It is a straight line that just "touches" a curve at a given point. This line represents the instantaneous direction of the curve, sharing the same slope at that very point. Think of it as the path the curve is following right at that instant, without capturing its twists.

To find a tangent line:

• Derivatives come into play. The first derivative of the curve at a point provides the slope of the tangent.
• For a point \((X_0, Y_0)\) on curve \(F\), the tangent line could be described using the formula \(Y - Y_0 = F'(X_0)(X - X_0)\).

In algebraic geometry, tangent lines are essential because they help us understand and visualize the behavior of a curve around any given point, especially when determining points like simple points and flex points.

Order of Contact

The "order of contact" is a bit like measuring how intimately a line and a curve are hugging one another at a point. It's a mathematical concept that describes how many times the line and the curve touch or coincide as they pass through that point. It's not just about them meeting once, but potentially touching in such a way that multiple derivatives align.

In practical terms, investigating the order of contact involves:

• Comparing differentials. When multiple derivatives of the curve and line match up at a point, this indicates higher contact orders.
• A higher order of contact means a higher degree of tangency. For instance, if \(L\)’s contact with \(F\) at \(P\) is third order, then \(L\) is more aligned with the curve than simply just touching it (i.e., first order).

This concept is important in defining flex points because it distinguishes between ordinary contact (simple intersection) and more profound intersections (like flex points of higher order). Understanding the order of contact helps in discerning the type and nature of these intersections.