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	ABOUT: [Introduction][History][Related areas][Subfields] POINTERS: [Texts][Software][Web links][Selected topics here] 26C: Polynomials, rational functions

Introduction

This page is reserved for the analytic study of polynomial functions. A great deal of information is available concerning the algebraic nature of polynomials, particularly those with integral (or rational) coefficients; this material is most in the section on Field Extensions; there is, there, a short guide to the placement of polynomial information across the Math Subject Classifications.

History

Applications and related fields

Comments about roots of polynomials (for example) which apply to more general functions than polynomials are considered with real functions. The use of polynomials for interpolation. The computational process of finding the roots is a task of numerical analysis. Families of orthogonal polynomials (e.g. Legendre polynomials) are treated as part of special functions and Fourier analysis

Subfields

• 26C05: Polynomials: analytic properties, etc., See also 12Dxx, 12Exx
• 26C10: Polynomials: location of zeros, See also 12D10, 30C15, 65H05
• 26C15: Rational functions, See also 14Pxx
• 26C99: None of the above but in this section

Parent field: 26: Real Functions.

Browse all (old) classifications for this area at the AMS.

Textbooks, reference works, and tutorials

Software and tables

Other web sites with this focus

Selected topics at this site

• Proofs of Descartes' Rule of Signs
• Sturm sequences - a technique to determine how many real roots a poly has.
• Using Sturm sequences to count real roots [in an interval]
• Citations for treatments of Sturm sequences and other methods of root-finding.
• What algorithm does (should?) Maple use for Sturm sequences?
• Use of Sturm sequences to determine if two ellipses intersect (without actually finding intersections!)
• One makes heavy use of symmetric polynomials. Here's an application to solving a system of (trigonometric) equations
• Finding a polynomial whose roots are products of the roots of two other polynomials.
• Old methods of root-finding: Graffe's, Vincent's.
• Under what conditions do all roots have magnitude 1?
• Polynomial with all roots on unit circle
• Criteria for roots of a polynomial to be outside the unit disc.
• Routh-Hurwitz criterion for all roots of a polynomial to have positive real part.
• Routh-Hurwitz criterion for all roots of a polynomial to be in the unit disk (left half-plane)
• What is the discriminant? (It's used to find multiple roots)
• Testing whether polynomials have only simple roots
• There is a classic but complicated formula for describing the roots of a general cubic equation. This has unusual behaviour when all three roots are real -- the so-called casus irreducibilis
• All the standard solutions to the cubic
• What is a Tschirnhaus transformation of a polynomial?
• How to reduce a quartic to simpler form (Möbius or Tschirnhaus transformation)
• Outline of procedures for solving a general quartic.
• Pointer: using elliptic functions to solve the quintic. [For the impossibility of solving the quintic with radicals only see 12F.]
• Closed-form (non-radical) solutions of quintic
• Theta functions: Sum(x^(n^2)), Jacobi identity, use to solve polynomial equations
• Sums of squares of real polynomials (Hilbert's 17th problem)
• Checking to see whether a (rational) polynomial is a sum of real squares.
• Polynomials which are positive across the real line
• Can one decompose polynomials into p = q o s + r with r small? (no).
• Characterizing polynomials by the pointwise vanishing of a high-order derivative
• The Fundamental Theorem of Algebra: every polynomial with complex coefficients has a (complex) root.
• Probability all roots of a random polynomial are real
• Methods of describing roots of high-order polynomials; decision procedure to decide whether distinct