Polynomials afg From Wikipedia, the free encyclopedia Contents 1 Abel polynomials 1 1.1 Examples ............................................... 1 1.2 References ............................................... 1 1.3 External links ............................................. 2 2 Abel–Ruffini theorem 3 2.1 Interpretation ............................................. 3 2.2 Lower-degree polynomials ....................................... 3 2.3 Quintics and higher .......................................... 4 2.4 Proof ................................................. 4 2.5 History ................................................ 5 2.6 See also ................................................ 6 2.7 Notes ................................................. 6 2.8 References .............................................. 6 2.9 Further reading ............................................ 6 2.10 External links ............................................. 6 3 Actuarial polynomials 7 3.1 See also ................................................ 7 3.2 References ............................................... 7 4 Additive polynomial 8 4.1 Definition ............................................... 8 4.2 Examples ............................................... 8 4.3 The ring of additive polynomials ................................... 9 4.4 The fundamental theorem of additive polynomials .......................... 9 4.5 See also ................................................ 9 4.6 References ............................................... 9 4.7 External links ............................................. 9 5 Alexander polynomial 10 5.1 Definition ............................................... 10 5.2 Computing the polynomial ...................................... 10 5.3 Basic properties of the polynomial .................................. 11 i ii CONTENTS 5.4 Geometric significance of the polynomial ............................... 11 5.5 Relations to satellite operations .................................... 12 5.6 Alexander–Conway polynomial .................................... 12 5.7 Relation to Khovanov homology ................................... 13 5.8 Notes ................................................. 13 5.9 References ............................................... 13 5.10 External links ............................................. 13 6 Algebraic equation 14 6.1 History ................................................ 14 6.2 Areas of study ............................................ 15 6.3 See also ................................................ 15 6.4 References ............................................... 16 7 Algebraic variety 17 7.1 Introduction and definitions ...................................... 18 7.1.1 Affine varieties ........................................ 18 7.1.2 Projective varieties and quasi-projective varieties ....................... 19 7.1.3 Abstract varieties ....................................... 19 7.2 Examples ............................................... 19 7.2.1 Subvariety ........................................... 20 7.2.2 Affine variety ......................................... 20 7.2.3 Projective variety ....................................... 21 7.3 Basic results .............................................. 22 7.4 Isomorphism of algebraic varieties .................................. 22 7.5 Discussion and generalizations ..................................... 22 7.6 Algebraic manifolds .......................................... 23 7.7 See also ................................................ 23 7.8 Footnotes ............................................... 24 7.9 References ............................................... 24 8 All one polynomial 25 8.1 Definition ............................................... 25 8.2 Properties ............................................... 25 8.3 References ............................................... 26 8.4 External links ............................................. 26 9 Almost linear hash function 27 9.1 References .............................................. 28 10 Alternating polynomial 29 10.1 Relation to symmetric polynomials .................................. 29 10.2 Vandermonde polynomial ....................................... 29 CONTENTS iii 10.2.1 Ring structure ......................................... 30 10.3 Representation theory ......................................... 30 10.4 Unstable ................................................ 31 10.5 See also ................................................ 31 10.6 Notes ................................................. 31 10.7 References ............................................... 31 11 Angelescu polynomials 32 11.1 See also ................................................ 32 11.2 References ............................................... 32 12 Appell sequence 33 12.1 Equivalent characterizations of Appell sequences ........................... 33 12.2 Recursion formula ........................................... 34 12.3 Subgroup of the Sheffer polynomials ................................. 34 12.4 Different convention .......................................... 35 12.5 See also ................................................ 35 12.6 References ............................................... 35 12.7 External links ............................................. 36 13 Bell polynomials 37 13.1 Complete Bell polynomials ...................................... 37 13.2 Combinatorial meaning ........................................ 38 13.2.1 Examples ........................................... 38 13.3 Properties ............................................... 38 13.3.1 Stirling numbers and Bell numbers .............................. 39 13.3.2 Touchard polynomials ..................................... 39 13.3.3 Convolution identity ..................................... 39 13.4 Applications of Bell polynomials ................................... 40 13.4.1 Faà di Bruno’s formula .................................... 40 13.4.2 Moments and cumulants ................................... 40 13.4.3 Representation of polynomial sequences of binomial type .................. 40 13.5 Software ................................................ 41 13.6 See also ................................................ 41 13.7 References ............................................... 41 14 Bernoulli polynomials 43 14.1 Representations ............................................ 44 14.1.1 Explicit formula ........................................ 44 14.1.2 Generating functions ..................................... 44 14.1.3 Representation by a differential operator ........................... 44 14.1.4 Representation by an integral operator ............................ 44 14.2 Another explicit formula ........................................ 45 iv CONTENTS 14.3 Sums of pth powers .......................................... 46 14.4 The Bernoulli and Euler numbers ................................... 46 14.5 Explicit expressions for low degrees .................................. 46 14.6 Maximum and minimum ....................................... 47 14.7 Differences and derivatives ...................................... 47 14.7.1 Translations .......................................... 48 14.7.2 Symmetries .......................................... 48 14.8 Fourier series ............................................. 48 14.9 Inversion ................................................ 49 14.10Relation to falling factorial ...................................... 50 14.11Multiplication theorems ........................................ 50 14.12Integrals ................................................ 50 14.13Periodic Bernoulli polynomials .................................... 51 14.14See also ................................................ 51 14.15References ............................................... 51 15 Bernstein polynomial 53 15.1 Definition ............................................... 54 15.2 Example ................................................ 54 15.3 Properties ............................................... 54 15.4 Approximating continuous functions ................................. 55 15.4.1 Proof ............................................. 56 15.5 See also ................................................ 56 15.6 Notes ................................................. 57 15.7 References ............................................... 57 15.8 External links ............................................. 57 16 Bernstein–Sato polynomial 58 16.1 Definition and properties ....................................... 58 16.2 Examples ............................................... 58 16.3 Applications .............................................. 59 16.4 References .............................................. 60 17 Binomial 61 17.1 Definition ............................................... 61 17.2 Operations on simple binomials .................................... 61 17.3 See also ................................................ 62 17.4 Notes ................................................. 62 17.5 References ............................................... 62 18 Boas–Buck polynomials 63 18.1 References ............................................... 63 CONTENTS v 19 Bollobás–Riordan polynomial 64 19.1 History ................................................. 64 19.2 Formal definition ........................................... 64 19.3 See also ................................................ 64 19.4 References ..............................................