Question

Let \(f\) be a rational function on a variety \(V\). Let \(U=\\{P \in V \mid f\) is defined at P\\}. Then \(f\) defines a function from \(U\) to \(k\). Show that this function determines \(f\) uniquely. So a rational function may be considered as a type of function, but only on the complement of an algebraic subset of \(V\), not on \(V\) itself.

Short answer

Based on the given step-by-step solution, the short answer is: If two rational functions, \(f\) and \(g\), agree on a set \(U \cap W\), which is the complement of an algebraic subset of a variety \(V\), then they must be the same function. This is proven by assuming that \(f \ne g\) and deriving a contradiction, which shows that the rational function is uniquely determined by its values on the complement of an algebraic subset of \(V\).

Step-by-step solution

Define rational functions

Let \(f\) and \(g\) be two distinct rational functions on the variety \(V\). Let \(U = \{P \in V \mid f\) is defined at \(P\}\) and \(W = \{P \in V \mid g\) is defined at \(P\}\). Assume that \(f(P) = g(P)\) for all \(P \in U \cap W\). We want to show that \(f = g\).

Define the difference of the rational functions

Let \(h(x) = f(x) - g(x)\) be the difference of the rational functions \(f\) and \(g\). Note that if \(h(x)\) is identically zero, then \(f(x) = g(x)\) for all \(x\) in the domain of \(f\) and \(g\), thus \(f = g\).

Analyze the zeros of \(h\)

For the points in \(U \cap W\), we have \(f(P) = g(P)\), so \(h(P) = 0\). Since \(h(x)\) is a rational function, it can only have finitely many zeros or it must be identically zero. We already assumed that \(f \ne g\), so \(h(x)\) cannot be identically zero. Thus, \(h(x)\) must have finitely many zeros.

Reach the contradiction

However, the definition of \(U \cap W\) implies that both \(f\) and \(g\) are defined at all points in \(U \cap W\), and the variety \(V\) is infinite. Thus, we have that \(h(x)\) agrees with the zero function on an infinite set (\(U \cap W\)), while having only finitely many zeros. This is a contradiction, and therefore our assumption that \(f \ne g\) is incorrect.

Conclusion

Since we have reached a contradiction, it follows that if \(f(P) = g(P)\) for all \(P \in U \cap W\), then \(f = g\). Therefore, a rational function is uniquely determined by its values on the complement of an algebraic subset of \(V\).

Key concepts

Algebraic Geometry

Algebraic geometry is a fascinating branch of mathematics that studies geometrical properties and spatial relations through algebra. It primarily revolves around the exploration of solutions to systems of polynomial equations. These solutions, or "zero-sets," form intriguing shapes and structures known as algebraic varieties.
These varieties can range from simple lines and curves to more complex surfaces and higher-dimensional shapes. They serve as the central focus in algebraic geometry. By connecting algebra and geometry, this field bridges the gap between abstract equations and visual geometrics.
If one can understand the algebraic equations, then they can potentially infer the geometric shape and vice versa. This ability makes algebraic geometry a critical tool in fields that model the real world using mathematical representations.

Algebraic Subsets

Algebraic subsets are fundamental building blocks in algebraic geometry. They are essentially the solution sets of polynomial equations restricted to a particular domain. In simple terms, an algebraic subset is where a given polynomial evaluates to zero.
For example, on a variety or space defined by one or more equations, algebraic subsets are the "excised" pieces where a polynomial equation holds true. They play a crucial role in defining rational functions and understanding their domains.

• An algebraic subset is closed under intersection and finite union, which means combining them in these ways gives another algebraic subset.
• They help in defining the domain of rational functions as these functions are not defined on the entirety of a variety, rather on the complement of an algebraic subset.

These subsets, thus act as the regions where the dynamics of a function like zeros and poles become significant.

Varieties

In algebraic geometry, a variety is a central concept referring to the specific sets of solutions to systems of polynomial equations. Varieties embody the concept of solving equations graphically rather than numerically or algebraically, offering a unified approach for studying the geometrical properties of these solutions.
Varieties can be purely algebraic in nature, but they often represent geometric objects and surfaces as seen in geometrical figures, enabling visualization of algebraic abstractions.

• A variety can have multiple dimensions, for example, a curve is a one-dimensional variety while a surface is two-dimensional.
• Varieties are often equipped with additional structure or parameters that describe more complex geometrical or topological properties.

By studying varieties, one gains insights into the symmetries and structures inherent to polynomial equations themselves.

Uniqueness

Uniqueness in mathematics, especially in the context of algebraic geometry, refers to the distinctly defined output of a function or a solution to an equation for a given input or within a specific condition set. This uniqueness property ensures that certain behaviors and functionalities are consistently preserved.
In the context of rational functions involving varieties, uniqueness means that if you know a function on a large enough domain or set, it can uniquely determine the function throughout that domain. This is exactly what was demonstrated in the exercise where it was shown that a rational function, defined at all points in the complement of an algebraic subset of a variety, can determine itself uniquely.

• This principle guarantees that two potentially different rational functions are indeed identical if they coincide over their entire domain.
• The consistency and predictability brought by uniqueness provide the cornerstone upon which many proofs and principles in algebraic geometry are constructed.

This is why showing the function's uniqueness is often a crucial step in proving properties and relationships in algebraic geometry.