Poincarépolynoom

A Poincaré polynomial is a mathematical object used in algebraic topology and algebraic geometry to study the structure of topological spaces and algebraic varieties. It is a polynomial whose coefficients encode the Betti numbers of the space, which are fundamental invariants that measure the number of "holes" of different dimensions. For a topological space X, its Poincaré polynomial P_X(t) is defined as the sum of t raised to the power of the Betti number b_i for each dimension i: P_X(t) = b_0 + b_1 t + b_2 t^2 + ... . The Betti numbers are typically non-zero only up to a certain dimension, making the polynomial finite.

The Poincaré polynomial provides a concise way to represent the homological information of a space. For example, a sphere has a simple Poincaré polynomial, reflecting its basic connectivity. More complex spaces will have more intricate polynomials. In the context of algebraic geometry, the Poincaré polynomial is also studied for smooth projective varieties. It is closely related to other important invariants such as the Hodge numbers, and for certain types of varieties, these polynomials can be computed or constrained by their geometric properties. The concept is named after the mathematician Henri Poincaré.