Mathematics and statistics

Pólya conjecture

The Pólya conjecture is a 1919 statement in number theory asserting the summatory Liouville function is never positive. Brian Haselgrove disproved it in 1958, and the smallest counterexample, 906,150,257, was found in 1980.

Pólya enumeration theorem

The Pólya enumeration theorem, or Pólya enumeration formula, is a combinatorics result counting distinct colorings and configurations up to a symmetry group, generalizing Burnside's lemma through weighted generating functions.

Pólya tree prior

The Pólya tree prior is a Bayesian nonparametric prior on distribution functions, built by recursive binary splitting with independent Beta branching probabilities, used in density estimation and survival analysis.

Pólya urn model

The Pólya urn model, or Pólya's urn, is a stochastic process where drawn balls are returned with extra balls of the same color, reinforcing frequent colors.

Polycube

A polycube is a solid figure formed by joining unit cubes face to face, the three-dimensional generalization of the two-dimensional polyomino, introduced by Martin Gardner in 1969.

Polygon

A polygon, also called an N-gon, is a closed plane figure bounded by straight segments called sides; the interior angles of an n-sided polygon sum to (n-2) × 180 degrees.

Polygon mesh

A polygon mesh is a collection of vertices, edges, and faces that defines the surface of a 3D object, the standard representation in computer graphics and games.

Polygon triangulation

Polygon triangulation, also known as polygon division, is the decomposition of a simple polygon into triangles with non-intersecting interiors, a basic primitive in computer graphics and computational geometry.

Polyhedron

A polyhedron (plural: polyhedra) is a three-dimensional solid bounded by flat polygonal faces joined along straight edges, with corners called vertices; examples include the cube, pyramid, and prism.

Polylogarithm

The polylogarithm, also called Jonquière's function, is a special function defined by the series Li_s(z) = sum z^n/n^s, generalizing the logarithm and Riemann zeta function.

Polymake

Polymake is open source software for research in polyhedral geometry, developed since 1997 at TU Berlin by Ewgenij Gawrilow and Michael Joswig, handling polytopes, fans, matroids, and tropical hypersurfaces.

Polymath Project

The Polymath Project, whose papers are signed D. H. J. Polymath, is a series of online collaborative mathematics projects launched by Timothy Gowers in January 2009 through blog comments.

Polymatroid

A polymatroid is a combinatorial structure in matroid theory, defined either as a polytope or by a monotone, submodular rank function; Jack Edmonds introduced it in 1970 as a generalization of the matroid.

Polynomial

A polynomial is a mathematical expression built from powers of one or more variables multiplied by coefficients, forming the basic object of algebra and a key tool in cryptography and numerical analysis.

Polynomial chaos

Polynomial chaos is an uncertainty quantification method that represents a random model output as a series of orthogonal polynomials of the input random variables, originating in Norbert Wiener's 1938 work.

Polynomial chaos expansion

Polynomial chaos expansion (PCE) is an uncertainty quantification method that represents a model's output as a series of orthogonal polynomials in random inputs, introduced by Norbert Wiener in 1938.

Polynomial hierarchy

The polynomial hierarchy (PH) is a tower of complexity classes between P and PSPACE whose levels use a constant number of alternating quantifiers; whether it is infinite remains open.

Polynomial long division

Polynomial long division, also known as the polynomial division algorithm, divides one polynomial by another to produce a quotient and a remainder, like arithmetic long division.

Polynomial method

The polynomial method is a collection of techniques in combinatorics and number theory that controls finite sets of points with suitably chosen polynomials, emerging in the late 2000s.

Polynomial regression

Polynomial regression is a form of regression analysis that models a response as a polynomial in the predictors, fitted by least squares like ordinary linear regression.

Polynomial ring

A polynomial ring, also called a polynomial algebra, is a ring whose elements are polynomials in one or more indeterminates, written R[x] over a base ring R.

Polynomial Szemerédi theorem

The polynomial Szemerédi theorem is a result in additive combinatorics, proved by Bergelson and Leibman in 1996, extending Szemerédi's theorem to polynomial patterns in dense integer sets.

Polynomial transformation (statistics)

A polynomial transformation adds powers and cross-products of existing predictors as new features, so a linear regression model can represent curved relationships while remaining linear in its coefficients.

Polynomial-time approximation scheme

A polynomial-time approximation scheme (PTAS) is an algorithmic framework for NP-hard optimization problems that, given an accuracy parameter ε, returns a solution within a factor of the optimum in polynomial time.

Polytope

A polytope is a geometric object bounded by lines, planes, or hyperplanes: a polygon in two dimensions, a polyhedron in three, and a bounded region of n-dimensional space in general.

Pons asinorum

Pons asinorum, or "bridge of asses," is the traditional name of the isosceles triangle theorem, Proposition 5 of Book I of Euclid's Elements: in an isosceles triangle the base angles are equal.

Pontryagin's maximum principle

Pontryagin's maximum principle is a set of necessary conditions for optimality in optimal control theory, announced in 1956 by Lev Pontryagin and students at the Steklov Mathematical Institute.

Pooled analysis

Pooled analysis combines the individual participant data of multiple studies into a single dataset, estimating an effect with greater precision; the two-stage approach, which preserves study membership, is the most common method.

Pooled variance

Pooled variance, also called combined variance, is a single estimate of a common population variance that weights group sample variances by degrees of freedom, used in the equal-variance two-sample t-test and analysis of variance.