polynomial

polynomial
Name	polynomial
Field	Isaac Newton, Carl Friedrich Gauss, Évariste Galois
Introduced	Renaissance mathematics, Ancient Greece
Notable examples	Pythagorean theorem, Binomial theorem, Taylor series

polynomial

A polynomial is an expression formed from variables and coefficients using addition, subtraction, multiplication, and nonnegative integer exponents; it plays a central role in Isaac Newton's work, Carl Friedrich Gauss's theorems, and Évariste Galois's theory. Polynomials underpin techniques in Leonhard Euler's analysis, Joseph-Louis Lagrange's interpolation, and algorithms used by Alan Turing-era computation and modern John von Neumann architectures. They connect classical results like the Binomial theorem and modern topics such as Fourier transform methods and Hilbert space techniques.

Definition and notation

A polynomial in one variable over a ring or field is typically written with coefficients and powers as a_0 + a_1 x + ... + a_n x^n; notation and formal definitions appear in works by Augustin-Louis Cauchy, Niels Henrik Abel, and David Hilbert. Standard symbols include variables like x, y, z and coefficient domains such as integers studied by Pierre de Fermat, rationals examined by Leonhard Euler, reals formalized by Bernhard Riemann, and complex numbers developed by William Rowan Hamilton. Degree, leading coefficient, and zero polynomial terminology derive from naming conventions used in texts by Emmy Noether and Richard Dedekind. Polynomial rings and modules are formalized in treatises by Emmy Noether and Saunders Mac Lane.

Algebraic properties

Polynomials satisfy closure properties under operations studied by Niels Henrik Abel and Évariste Galois and obey unique factorization in principal ideal domains like Gauss's work on integers and Richard Dedekind's ideals. Concepts of reducibility, irreducibility, and greatest common divisors are treated in the context of Euclid's algorithm, refined by Srinivasa Ramanujan and Carl Friedrich Gauss. The structure of polynomial rings over fields is central to Emmy Noether's abstract algebra, while modules and homomorphisms connect to results in Hermann Weyl's representation theory and David Hilbert's basis theorems.

Operations (addition, multiplication, division)

Addition and multiplication of polynomials follow distributive and associative laws formalized by Blaise Pascal and demonstrated in Binomial theorem proofs credited to Isaac Newton and Blaise Pascal. Polynomial long division and synthetic division techniques trace to algorithms used by Joseph-Louis Lagrange and present-day computational implementations in systems inspired by Alan Turing and John von Neumann. Euclidean division yields quotient and remainder concepts foundational to Carl Friedrich Gauss's number-theoretic results and to modern symbolic computation as implemented in software developed by teams at MIT and Stanford University.

Roots, factorization, and the Fundamental Theorem of Algebra

The study of roots connects to classical problems addressed by Girolamo Cardano and to Évariste Galois's criteria for solvability by radicals; the Fundamental Theorem of Algebra, proved in rigorous form by Carl Friedrich Gauss, guarantees at least one complex root for nonconstant polynomials over ℂ. Factorization over various fields—rationals studied by Pierre de Fermat, reals by Augustin-Louis Cauchy, and finite fields used in Claude Shannon-inspired coding theory—leads to distinct behavior explored in Évariste Galois's group theory and Richard Dedekind's ideal theory. Multiplicity, algebraic conjugates, and splitting fields are central in the work of Emmy Noether, David Hilbert, and Emil Artin.

Special classes of polynomials

Several named families play prominent roles: Chebyshev polynomials used in approximation theorists' work linked to Pafnuty Chebyshev and Andrey Markov; Legendre and Hermite polynomials tied to Adrien-Marie Legendre and Charles Hermite in mathematical physics; orthogonal polynomials appearing in John von Neumann-inspired quantum mechanics; cyclotomic polynomials central to Gauss's construction problems; and minimal polynomials studied by Évariste Galois and Emil Artin. Bernstein polynomials appear in Sergei Bernstein's approximation results; Bézout polynomials relate to Étienne Bézout's elimination theory; and Sturm polynomials derive from work of Charles-François Sturm.

Applications and connections to other fields

Polynomials are applied across diverse domains: interpolation and approximation in numerical analysis advanced by Joseph-Louis Lagrange and Carl Runge; signal processing and Fourier methods used in Norbert Wiener's work; control theory relying on characteristic polynomials in results associated with Rudolf Kalman; cryptography and coding theory using polynomials over finite fields in systems developed at Bell Labs and by researchers like Claude Shannon; algebraic geometry linking to schemes and sheaf theory in the work of Alexander Grothendieck and Jean-Pierre Serre; and computational complexity theory studied by Stephen Cook and Leslie Valiant via polynomial-time classes. Engineering, physics, and economics modeling often use polynomial approximations popularized by approximation theory texts and applications in NASA projects and European Space Agency missions.

Category:Algebra