Question

Let \(F \in k\left[X_{1}, \ldots, X_{n+1}\right]\left(k\right.\) infinite). Write \(F=\sum F_{i}, F_{i}\) a form of degree \(i .\) Let \(P \in \mathbb{P}^{n}(k)\), and suppose \(F\left(x_{1}, \ldots, x_{n+1}\right)=0\) for every choice of homogeneous coordinates \(\left(x_{1}, \ldots, x_{n+1}\right)\) for \(P .\) Show that each \(F_{i}\left(x_{1}, \ldots, x_{n+1}\right)=0\) for all homogeneous coordinates for P. (Hint: consider \(G(\lambda)=F\left(\lambda x_{1}, \ldots, \lambda x_{n+1}\right)=\sum \lambda^{i} F_{i}\left(x_{1}, \ldots, x_{n+1}\right)\) for fixed \(\left.\left(x_{1}, \ldots, x_{n+1}\right) .\right)\)

Short answer

Question: Show that if a polynomial F is zero on a given point P in the projective space for all homogeneous coordinates, then each of its homogeneous components F_i is also zero on P. Answer: We introduced a new polynomial G(λ) and evaluated it at the point P for all λ. We found that G(λ) has an infinite number of roots, which implies that it is the zero polynomial. This means that all of its coefficients must be zero, and hence each homogeneous part of G, λ^i F_i(x_1, \ldots, x_{n+1}), must be zero. Therefore, F_i(x_1, \ldots, x_{n+1}) = 0 for all homogeneous coordinates for P and for each i, which completes the proof.

Step-by-step solution

Introduce the polynomial G

Define a new polynomial G(λ) such that $$ G(\lambda) = F(\lambda x_1, \ldots, \lambda x_{n+1}) = \sum \lambda^i F_i(x_1, \ldots, x_{n+1}). $$ Since F is a homogeneous polynomial, G is a polynomial in λ.

Evaluate G(λ) at P

We are given that F is zero on P for all homogeneous coordinates \((x_1,\dots,x_{n+1})\), so we can evaluate G(λ) at P: $$ G(\lambda) = F(\lambda x_1, \ldots, \lambda x_{n+1}) = 0, $$ where the last equality holds for every choice of homogeneous coordinates for P and for all λ.

Analyze the roots of G(λ)

Since G(λ) is a polynomial in λ and it is zero for all λ, this means that G(λ) has an infinite number of roots. But the only way a polynomial can have an infinite number of roots is if it is the zero polynomial. So, we have that $$ G(\lambda) = \sum \lambda^i F_i(x_1, \ldots, x_{n+1}) = 0 $$ for all λ and all homogeneous coordinates \((x_1,\dots,x_{n+1})\) for P.

Show that each F_i is zero on P

We know that G(λ) is the zero polynomial, so all of its coefficients must be zero. This means that each homogeneous part of G, λ^i F_i(x_1, \ldots, x_{n+1}), must be zero. Hence, we have $$ F_i(x_1, \ldots, x_{n+1}) = 0 $$ for all homogeneous coordinates \((x_1,\dots,x_{n+1})\) for P and for each i. This completes the proof that if F is zero on P for all homogeneous coordinates, then each of its homogeneous components F_i is also zero on P.

Key concepts

Algebraic Curves

In the world of mathematics, specifically within algebraic geometry, algebraic curves hold significant interest due to their intrinsic beauty and complexity. Imagine an algebraic curve as a one-dimensional geometric figure, not unlike the curves we see in everyday life, but defined in a very particular way. These curves are represented by equations of the form F(x, y) = 0, where F is a polynomial in two variables x and y with coefficients from a field, usually denoted by k.

Algebraic curves can be neither too 'curvy' nor too 'straight'; they inhabit a world where 'straight lines' are generalizations that extend to infinity in both directions and 'curves' may close in on themselves to form loops. A key aspect of studying such curves is to investigate their properties, understand their intersections, and classify them into types based on features such as singularity and genus.

Exploring the case of the homogeneous polynomial F from our problem, we witness a powerful property of algebraic curves: when 'cut' by an algebraic curve, the zeros of F form a set that is invariant with respect to scaling of coordinates. This behavior is precisely what underpins the proof in our exercise, illustrating that if the polynomial is zero for one set of homogeneous coordinates at a point P, it remains zero even if we 'stretch' or 'shrink' those coordinates proportionally.

Algebraic Geometry

Delve deeper, and we enter the realm of algebraic geometry, which is a fusion of abstract algebra and geometry. This field profoundly explores the multi-dimensional analogues of algebraic curves, called algebraic varieties, of which curves are a special case. These varieties are sets of solutions to systems of polynomial equations containing several variables.

In the case of the exercise dealing with the homogeneous polynomial F and its relationship with a point P in projective space, algebraic geometry provides the theoretical backdrop. By using concepts from algebraic geometry, such as varieties and notions of dimensionality, we gain the understanding necessary to piece apart and delve into the nature of the polynomial's zero set.

The methodical breakup of F into homogeneous components F_i aligns with the algebraic geometrical approach of studying objects through their 'slices' and 'projections'—simpler pieces that can be more easily understood and manipulated. When we show that each F_i must be zero if F is zero, we borrow from the algebraic geometry toolkit that allows us to infer the properties of the entire object from its parts, much akin to solving a puzzle.

Projective Space

In the universe of higher mathematics, the idea of projective space, denoted by \(\text{ℙ}^n(k)\), provides us with a way to look at geometry without the restrictions of 'perspective'. Essentially, projective space is an extension of the traditional two-dimensional plane and three-dimensional space into an 'n-dimensional' space, accounting for the concept of points at infinity.

Projective space can seem abstract, but one can think of it like a vast canvas where every line extends infinitely in both directions and intersects with a unique 'point at infinity'. Here, points are no longer defined by their usual (x, y) or (x, y, z) coordinates but by ratios of coordinates, allowing for 'homogeneous coordinates' which are not unique but proportional to one another.

The homogeneous nature of coordinates in projective space ties back neatly to our exercise. When F is zero for a certain point P, it remains consistently zero across all scaled versions of that point's coordinates. This invariance is crucial for proving that each component F_i individually vanishes at P. This property is specific to the projective space framework and sheds light on the 'continuous' and holistic attributes of algebraic objects under its lens.