Euclid’s Lemma

Bézout's theorem is a fundamental statement in intersection theory. It asserts that the number of intersection points of two plane algebraic curves of degrees \(m\) and \(n\) is exactly \(mn\), provided that one works in a projective plane over an algebraically closed field, counts points with multiplicity, and includes points at infinity where parallel asymptotes meet.

This theorem states that every integer greater than 1 is either a prime number or can be uniquely represented as a product of prime numbers, disregarding the order of the factors. For example, \(1200 = 2^4 \times 3^1 \times 5^2\). This unique factorization is a cornerstone of number theory, providing a fundamental multiplicative structure for the integers.

Hilbert's Nullstellensatz (German for "theorem of zeros") establishes a fundamental correspondence between geometry and algebra. It states that for an algebraically closed field \(k\), if a polynomial \(p\) vanishes on the zero-set of an ideal \(I\), then some power of \(p\) must belong to \(I\). Formally, \(I(V(I)) = \sqrt{I}\), the radical of \(I\).

Sheaf cohomology is a central tool in modern algebraic geometry for studying global properties of geometric spaces. For a sheaf \(\mathcal{F}\) on a space \(X\), the cohomology groups \(H^i(X, \mathcal{F})\) are vector spaces whose dimensions provide important invariants. The group \(H^0\) represents global sections, while higher groups \(H^i\) for \(i > 0\) measure the obstructions to patching together local sections into a global one.

A partial differential equation (PDE) is an equation that imposes relations between the various partial derivatives of a multivariable function. The function is often called the unknown, and a PDE describes a relationship between this unknown function and its derivatives. Unlike ordinary differential equations (ODEs), which involve functions of a single variable, PDEs are fundamental to modeling multidimensional systems.

A technique for solving first-order and hyperbolic second-order partial differential equations (PDE). The method reduces a PDE to a family of ordinary differential equations (ODEs) along specific curves called 'characteristics'. Along these curves, the PDE simplifies, allowing the solution to be found by integrating the system of ODEs. It is particularly powerful for problems involving transport and wave propagation.

A fundamental existence and uniqueness theorem for partial differential equations associated with Cauchy initial value problems. It states that if the PDE and the initial conditions are 'analytic' (can be represented by convergent power series), then a unique analytic solution exists in a neighborhood of the initial surface. It provides a local existence guarantee but does not address global behavior or well-posedness.

An affine variety is the set of points in an affine space whose coordinates are the common zeros of a finite set of polynomials. For a set of polynomials \(S = \{f_1, \dots, f_k\}\) in a polynomial ring \(k[x_1, \dots, x_n]\), the corresponding affine variety is \(V(S) = \{x \in k^n | f(x) = 0 \text{ for all } f \in S\}\). It is a central object of study in classical algebraic geometry.