Determine the number of triple points in a net of homogeneous polynomials on the projective line using first principles | Step-by-Step Solution

Understanding Triple Points in a Net on the Projective Line

What We're Solving: We need to find how many points on the projective line P¹ have the property that ALL polynomials in our net vanish to order at least 3 at those points. A net is a 3-dimensional family of polynomials, and we're looking for their common "triple points."

The Approach: We'll use the fact that a net gives us exactly the right number of constraints to determine isolated points, then count them using intersection theory and Bézout's theorem.

Step-by-Step Solution:

Step 1: Set up the problem systematically

• A net of degree d homogeneous polynomials on P¹ is a 3-dimensional linear system
• Each polynomial can be written as F(x,y) = Σaᵢxⁱy^(d-i) where i goes from 0 to d
• For degree 1: F(x,y) = ax + by, so our net is spanned by three linearly independent linear forms

Step 2: Understand what "vanishing to order ≥ 3" means

• At a point [x₀:y₀], a polynomial vanishes to order ≥ 3 if the polynomial, its first derivative, AND its second derivative all vanish there
• For degree 1 polynomials, this creates a contradiction! Here's why:

Step 3: Analyze the constraint mathematically

• A degree 1 polynomial F(x,y) = ax + by has derivative F'(x,y) = a (with respect to the affine coordinate x/y)
• If F vanishes to order ≥ 3, then F = 0, F' = 0, which means a = 0
• This forces F(x,y) = by, so F can only vanish at the single point [1:0]
• But then F doesn't vanish to order 3 there—it vanishes to order exactly 1!

Step 4: Apply the dimensional analysis

• For a polynomial to vanish to order ≥ 3 at a point, we need 3 conditions
• Our net is 3-dimensional, so these 3 conditions should generically determine isolated points
• However, degree 1 polynomials simply don't have enough "flexibility" to satisfy such strong vanishing conditions

Step 5: Conclude using the constraint counting Since no degree 1 polynomial can vanish to order ≥ 3 at any point (the derivative constraints are too restrictive for such low-degree polynomials), there are no points where an entire net of such polynomials can simultaneously satisfy this condition.

The Answer: The number of triple points is 0.

A net of degree 1 homogeneous polynomials on the projective line has no points where all polynomials in the net vanish to order ≥ 3.

Memory Tip: Remember that "low degree, high vanishing order" typically leads to contradictions! Degree 1 polynomials are too "simple" to have triple points. You need at least degree 3 polynomials to even have the possibility of triple points, since you need enough coefficients to satisfy all the vanishing conditions.

This problem beautifully illustrates how geometric constraints interact with the algebraic degrees of freedom available in your polynomial family!