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The Abel polynomials are a sequence of polynomials named after Niels Henrik Abel, defined by the following equation: p n ( x ) = x ( x − a n ) n − 1 {\displaystyle
Abel_polynomials
Equations of degree 5 or higher cannot be solved by radicals
finite number of polynomials, there are values of the variables at which none of the polynomials takes the value zero. Soon after Abel's publication of
Abel–Ruffini_theorem
Bernstein polynomial Characteristic polynomial Minimal polynomial Invariant polynomial Abel polynomials Actuarial polynomials Additive polynomials All one
List_of_polynomial_topics
other special polynomials, are included. Contents: Top 0–9 A B C D E F G H I J K L M N O P Q R S T U V W X Y Z Niels Abel: Abel polynomials - Abelian function
List of eponyms of special functions
List_of_eponyms_of_special_functions
Mathematical concept
composition of two polynomials is strongly related to the degree of the input polynomials. The degree of the sum (or difference) of two polynomials is less than
Degree_of_a_polynomial
Sequence valued in polynomials
All-one polynomials Abel polynomials Bell polynomials Bernoulli polynomials Cyclotomic polynomials Dickson polynomials Fibonacci polynomials Lagrange
Polynomial_sequence
Algorithm for division of polynomials
smaller ones. Polynomial long division is an algorithm that implements the Euclidean division of polynomials: starting from two polynomials A (the dividend)
Polynomial_long_division
Mathematical connection between field theory and group theory
introduced the subject for studying roots of polynomials. This allowed him to characterize the polynomial equations that are solvable by radicals in terms
Galois_theory
Algebraic structure
especially in the field of algebra, a polynomial ring or polynomial algebra is a ring formed from the set of polynomials in one or more indeterminates (traditionally
Polynomial_ring
Niels Henrik Abel in 1824, which made essential use of the Galois theory of field extensions. In the paper, Abel proved that polynomials with degree more
Polynomial_root-finding
Type of mathematical expression
polynomials, quadratic polynomials and cubic polynomials. For higher degrees, the specific names are not commonly used, although quartic polynomial (for
Polynomial
Mathematical approximation of a function
of a Taylor series is a polynomial of degree n that is called the nth Taylor polynomial of the function. Taylor polynomials are approximations of a function
Taylor_series
In mathematics, Abel–Goncharov interpolation determines a polynomial such that various higher derivatives are the same as those of a given function at
Abel–Goncharov_interpolation
theorem Abel polynomials Abel's summation formula Abelian means Abel's test Abel's theorem Abelian theorem Abel–Ruffini theorem Abel transform Abel transformation
List of things named after Niels Henrik Abel
List_of_things_named_after_Niels_Henrik_Abel
Norwegian mathematician (1802–1829)
Niels Henrik Abel (/ˈɑːbəl/ AH-bəl, Norwegian: [ˌnɪls ˈhɛ̀nːɾɪk ˈɑ̀ːbl̩]; 5 August 1802 – 6 April 1829) was a Norwegian mathematician who made pioneering
Niels_Henrik_Abel
Type of polynomial sequence
Abel polynomials The Bernoulli polynomials The Euler polynomials The central factorial polynomials The Hermite polynomials The Laguerre polynomials The
Sheffer_sequence
Polynomial equation, generally univariate
associated with the cyclotomic polynomials of degrees 5 and 17. Charles Hermite, on the other hand, showed that polynomials of degree 5 are solvable using
Algebraic_equation
Polynomial without nontrivial factorization
non-constant polynomials are exactly the polynomials that are non-invertible and non-zero. Another definition is frequently used, saying that a polynomial is irreducible
Irreducible_polynomial
Type of polynomial sequence
In mathematics, a polynomial sequence, i.e., a sequence of polynomials indexed by non-negative integers { 0 , 1 , 2 , 3 , … } {\textstyle \left\{0,1,2
Binomial_type
Polynomial function of degree 4
xi. By the fundamental theorem of symmetric polynomials, these coefficients may be expressed as polynomials in the coefficients of the monic quartic. If
Quartic_function
Mathematical formula involving a given set of operations
unit interval can be expressed as a limit of polynomials, so any class of functions containing the polynomials and closed under limits will necessarily include
Closed-form_expression
Roots of multiple multivariate polynomials
of polynomial equations (sometimes simply a polynomial system) is a set of simultaneous equations f1 = 0, ..., fh = 0 where the fi are polynomials in
System of polynomial equations
System_of_polynomial_equations
(Mathematical) decomposition into a product
factorizations within the ring of polynomials with rational number coefficients (see factorization of polynomials). A commutative ring possessing the
Factorization
Field theory result
If f(x) ≠ g(x) are two different irreducible monic polynomials, then they share no roots. Abel, N. H. (1829), "Mémoire sur une classe particulière d'équations
Abel's_irreducibility_theorem
Every polynomial has a real or complex root
non-constant single-variable polynomial with complex coefficients has at least one complex root. This includes polynomials with real coefficients, since
Fundamental theorem of algebra
Fundamental_theorem_of_algebra
Mathematical expression using basic operations
{1-x^{2}}{1+x^{2}}}}} An algebraic equation is an equation involving polynomials, for which algebraic expressions may be solutions. If the set of constants
Algebraic_expression
Polynomial function of degree 5
±2759640, in which cases the polynomial is reducible. As solving reducible quintic equations reduces immediately to solving polynomials of lower degree, only
Quintic_function
Generalization of elliptic integrals
an abelian integral, named after the Norwegian mathematician Niels Henrik Abel, is an integral in the complex plane of the form ∫ z 0 z R ( x , w ) d x
Abelian_integral
Branch of mathematics
above example). Polynomials of degree one are called linear polynomials. Linear algebra studies systems of linear polynomials. A polynomial is said to be
Algebra
Approximation of a function by a polynomial
Similarly, we might get still better approximations to f if we use polynomials of higher degree, since then we can match even more derivatives with
Taylor's_theorem
German mathematician (1804–1851)
of the first to introduce and study the symmetric polynomials that are now known as Schur polynomials, giving the so-called bialternant formula for these
Carl_Gustav_Jacob_Jacobi
polynomials Continuous q-Jacobi polynomials Big q-Jacobi polynomials Little q-Jacobi polynomials Pseudo Jacobi polynomials Sieved Jacobi polynomials Jacobi
List of things named after Carl Gustav Jacob Jacobi
List_of_things_named_after_Carl_Gustav_Jacob_Jacobi
Type of complex number
These numbers are roots of polynomials of degree 5 or higher, a result of Galois theory (see Quintic equations and the Abel–Ruffini theorem). For example
Algebraic_number
exponential map through the corresponding Abel function X {\displaystyle {\mathcal {X}}} , satisfying the related Abel equation X ( exp ( u ) ) = X ( u )
Superfunction
Algebraic structure with addition, multiplication, and division
the following polynomials are not expressible by sums, products, and radicals. For the latter polynomial, this fact is known as the Abel–Ruffini theorem:
Field_(mathematics)
Skeletonized version of algebraic geometry
In mathematics, tropical geometry is the study of polynomials and their geometric properties when addition is replaced with minimization and multiplication
Tropical_geometry
French mathematician (1752–1833)
mathematics. Well-known and important concepts such as the Legendre polynomials and Legendre transformation are named after him. He is also known for
Adrien-Marie_Legendre
Polynomial equation of degree 3
polynomials in r1, r2, r3, and a. The proof then results in the verification of the equality of two polynomials. If the coefficients of a polynomial are
Cubic_equation
Method of differentiating single-term polynomials
differentiation is a linear operation on the space of differentiable functions, polynomials can also be differentiated using this rule. The power rule underlies
Power_rule
Complex number that solves a monic polynomial with integer coefficients
is contained in the ideal generated by its two input polynomials.) Every root of a monic polynomial whose coefficients are algebraic integers is itself
Algebraic_integer
American mathematician and Nobel Laureate (1928–2015)
his work, Nash proved that those smooth functions can be taken to be polynomials. This was widely regarded as a surprising result, since the class of
John_Forbes_Nash_Jr.
Classical orthogonal polynomials Hermite polynomials Laguerre polynomials Jacobi polynomials Gegenbauer polynomials Legendre polynomials Euclidean space Metric
List_of_real_analysis_topics
Mathematical formula expressing equality
equation is a polynomial equation (commonly called also an algebraic equation) in which the two sides are polynomials. The sides of a polynomial equation contain
Equation
French mathematician (1822–1901)
In 1864, Hermite presented a new class of special functions, Hermite polynomials, in the context of expansions in terms of continuous functions over unbounded
Charles_Hermite
Study of polynomial equations
cannot be solved in radicals followed by Niels Henrik Abel's complete proof in 1824 (now known as the Abel–Ruffini theorem). Évariste Galois later introduced
Theory_of_equations
Infinite sum of monomials
can view power series as being like "polynomials of infinite degree", although power series are not polynomials in the strict sense. The geometric series
Power_series
Hungarian-born American mathematician (1926–2025)
May 1926 – 16 May 2025) was a Hungarian-born American mathematician and Abel Prize laureate working in the areas of pure and applied mathematics. Lax
Peter_Lax
Mathematical term; type of polynomial transformation
of mapping on polynomials developed by Ehrenfried Walther von Tschirnhaus in 1683. Simply, it is a method for transforming a polynomial equation of degree
Tschirnhaus_transformation
Italian mathematician and philosopher (1765–1822)
could not accept his revolutionary idea that a polynomial could not be solved in radicals. Niels Henrik Abel is sometimes incorrectly credited with Ruffini's
Paolo_Ruffini
Dimensionless quantity in fluid dynamics
be solved explicitly, the Abel–Ruffini theorem guarantees that there exists no general form for the roots of these polynomials). It is first determined
Mach_number
mathematician, known for Rogers–Askey–Ismail polynomials, Al-Salam–Ismail polynomials and Chihara–Ismail polynomials Peter Medawar, Lebanese-British biologist
List of modern Arab scientists and engineers
List_of_modern_Arab_scientists_and_engineers
Hungarian mathematician (born 1948)
best known for his work in combinatorics, for which he was awarded the 2021 Abel Prize jointly with Avi Wigderson. He was the president of the International
László_Lovász
Inverse of a finite difference
where B a ( x ) {\displaystyle B_{a}(x)} are the Bernoulli polynomials (via Abel-Plana, Hurwitz zeta, or as defined by their recurrence; not the
Indefinite_sum
Theorem in algebraic geometry
homogeneous polynomials in n variables is the value of a polynomial function of the coefficients, which takes the value zero if and only if the polynomials have
Main theorem of elimination theory
Main_theorem_of_elimination_theory
Mathematical operation
− m ≥ 0. Fourier transform Integral transform Abel transform Fourier–Bessel series Neumann polynomial Y and H transforms Louis de Branges (1968). Hilbert
Hankel_transform
Arithmetic operation, inverse of nth power
polynomial roots. The quadratic formula expresses the roots of quadratic polynomials in terms of square roots. During the 16th century, Gerolamo Cardano and
Nth_root
American mathematician (born 1934)
Mathematical Society Strang, Gilbert (November 1, 1973). "Piecewise polynomials and the finite element method". Bulletin of the American Mathematical
Gilbert_Strang
Japanese mathematician (born 1947)
thesis proves the rationality of the roots of b-functions (Bernstein–Sato polynomials), using D-module theory and resolution of singularities. Kashiwara's
Masaki_Kashiwara
Number with a real and an imaginary part
of all such polynomials is denoted by R [ X ] {\displaystyle \mathbb {R} [X]} . Since sums and products of polynomials are again polynomials, this set R
Complex_number
Norwegian mathematician (1899–1968)
defined, and the Ore extension, a non-commutative analogue of rings of polynomials, are part of this work. In more elementary number theory, Ore's harmonic
Øystein_Ore
French mathematician (born 1926)
number theory. He was awarded the Fields Medal in 1954 and the inaugural Abel Prize in 2003. Born in Bages, Pyrénées-Orientales, to pharmacist parents
Jean-Pierre_Serre
Analytic function that does not satisfy a polynomial equation
are algebraically independent of some class, such as when one takes polynomials with trigonometric functions as variables. Most familiar transcendental
Transcendental_function
Solution in radicals of a polynomial equation
quartic equations, which are more complicated than the quadratic formula. The Abel–Ruffini theorem, and, more generally Galois theory, state that some quintic
Solution_in_radicals
Fully simplified fraction
rational fractions such that the numerator and the denominator are coprime polynomials. Every rational number can be represented as an irreducible fraction
Irreducible_fraction
theorem (polynomials) Polynomial remainder theorem (polynomials) Primitive element theorem (field theory) Rational root theorem (algebra, polynomials) Solutions
List_of_theorems
Square of a triangular number
(2004) study polynomial analogues of the square triangular number formula, in which series of polynomials add to the square of another polynomial. Stroeker
Squared_triangular_number
Tauberian theorem
asymptotics of the partial sums of a series with the asymptotics of its Abel summation. In this form, the theorem asserts that if the sequence a n ≥ 0
Hardy–Littlewood Tauberian theorem
Hardy–Littlewood_Tauberian_theorem
Concepts from linear algebra
algebraic formulas for the roots of a polynomial exist only if the degree n is 4 or less. According to the Abel–Ruffini theorem there is no general, explicit
Eigenvalues_and_eigenvectors
Mapping involving integration between function spaces
Laplace transform, one obtains a time-domain solution. In this example, polynomials in the complex frequency domain (typically occurring in the denominator)
Integral_transform
Special mathematical function
ISBN 978-2-88124-682-1. (see § 1.2, "The generalized zeta function, Bernoulli polynomials, Euler polynomials, and polylogarithms", p. 23.) Robinson, J.E. (1951). "Note on
Polylogarithm
French mathematician (1811–1832)
the memoir was lost. The prize would be awarded that year to Niels Henrik Abel posthumously and also to Carl Gustav Jacob Jacobi. Despite the lost memoir
Évariste_Galois
Number of subsets of a given size
combination of binomial coefficient polynomials is integer-valued too. Conversely, (4) shows that any integer-valued polynomial is an integer linear combination
Binomial_coefficient
Projective variety that is also an algebraic group
and quartic polynomials. When those were replaced by polynomials of higher degree, say quintics, what would happen? In the work of Niels Abel and Carl Jacobi
Abelian_variety
Class of periodic mathematical functions
studied by Legendre, whose work was taken on by Niels Henrik Abel and Carl Gustav Jacobi. Abel discovered elliptic functions by taking the inverse function
Elliptic_function
Mathematics award
survey conducted by IREG in 2013–14, the Fields Medal came closely after the Abel Prize as the second most prestigious international award in mathematics.
Fields_Medal
Commutative group (mathematics)
mathematician Niels Henrik Abel, who had found that the commutativity of the group of a polynomial implies that the roots of the polynomial can be calculated by
Abelian_group
Polynomial equation of degree 4
this was the highest order general polynomial for which such solutions could be found was first given in the Abel–Ruffini theorem in 1824, proving that
Quartic_equation
History of a branch of mathematics
theory and geometry. Joseph Louis Lagrange, Paolo Ruffini, Niels Henrik Abel and Évariste Galois were early researchers in the field of group theory.
History_of_group_theory
Linear combination of nth roots
(nests) another radical expression Abel–Ruffini theorem states that there is no solution in radicals to general polynomial equations of degree five or higher
Sum_of_radicals
Infinite series with alternating signs
, so the series is an example where a slightly stronger method, such as Abel summation, is required. The series 1 − 2 + 3 − 4 + ... is closely related
1_−_2_+_3_−_4_+_⋯
Summation method for some divergent series
strictly weaker than Borel summation; for q > 0 they are incomparable with Abel summation. For some value y we may define the Euler sum (if it converges
Euler_summation
Russian-French mathematician
mathematics at New York University. Gromov has won several prizes, including the Abel Prize in 2009 "for his revolutionary contributions to geometry". Mikhail
Mikhael Gromov (mathematician)
Mikhael_Gromov_(mathematician)
Model of quantum computation
34th Annual Symposium on Foundations of Computer Science. pp. 352–361. Abel Molina; John Watrous (2018). "Revisiting the simulation of quantum Turing
Quantum_Turing_machine
Result of multiplying four instances of a number together
Fourth-degree equations, which contain a fourth degree (but no higher) polynomial are, by the Abel–Ruffini theorem, the highest degree equations having a general
Fourth_power
Branch of discrete mathematics
and is closely related to q-series, special functions and orthogonal polynomials. Originally a part of number theory and analysis, it is now considered
Combinatorics
Contents: Top 0–9 A B C D E F G H I J K L M N O P Q R S T U V W X Y Z Abel 1. Abel sum 2. Abel integral absolute absolute convergence accumulation An accumulation
Glossary of real and complex analysis
Glossary_of_real_and_complex_analysis
quartic polynomials. When those were replaced by polynomials of higher degree, say quintics, what would happen? In the work of Niels Henrik Abel and Carl
History of manifolds and varieties
History_of_manifolds_and_varieties
Mathematical technique for simplification
polynomial equations are generally impossible to solve in terms of radicals (see Abel–Ruffini theorem). This particular equation, however, may be written ( x 3
Change_of_variables
uniformly by polynomials, or certain other function spaces Approximation by polynomials: Linear approximation Bernstein polynomial — basis of polynomials useful
List of numerical analysis topics
List_of_numerical_analysis_topics
Relates theta constants to the branch points of a hyperelliptic curve
hyperelliptic curve (Mumford 1984, section 8). In 1824, the Abel–Ruffini theorem established that polynomial equations of a degree of five or higher could have
Thomae's_formula
Matrix decomposition
difficult to compute and express: the Abel–Ruffini theorem implies that the roots of high-degree (5 or above) polynomials cannot in general be expressed simply
Eigendecomposition of a matrix
Eigendecomposition_of_a_matrix
Numerical methods for matrix eigenvalue calculation
for finding eigenvalues could also be used to find the roots of polynomials. The Abel–Ruffini theorem shows that any such algorithm for dimensions greater
Eigenvalue_algorithm
Group with subnormal series where all factors are abelian
in the proof that for every n > 4 there are polynomials of degree n which are not solvable by radicals (Abel–Ruffini theorem). This property is also used
Solvable_group
Algebraic variety in a projective space
{\displaystyle \mathbb {P} ^{n}} of some finite family of homogeneous polynomials that generate a prime ideal, the defining ideal of the variety. A projective
Projective_variety
Russian mathematician (born 1943)
doi:10.1007/BF02465190 Ilyashenko, Yu (2000). "Hilbert-type numbers for Abel equations, growth and zeros of holomorphic functions". Nonlinearity. 13 (4):
Yulij_Ilyashenko
Mathematical series
applications of infinite series, Blackie and Son, §22. Coolidge 1949. Abel 1826. Abel, Niels (1826), "Recherches sur la série 1 + (m/1)x + (m(m − 1)/1.2)x2
Binomial_series
Mathematical field obtained by adjunction of nth roots
a splitting field of f over K contained in a radical extension of K. The Abel–Ruffini theorem states that such a solution by radicals does not exist, in
Radical_extension
Methods of calculating definite integrals
interpolating functions are polynomials. In practice, since polynomials of very high degree tend to oscillate wildly, only polynomials of low degree are used
Numerical_integration
Analytic function in mathematics
\varepsilon >0)} Peter Borwein developed an algorithm that applies Chebyshev polynomials to the Dirichlet eta function to produce a very rapidly convergent series
Riemann_zeta_function
1984 – Vaughan Jones discovers the Jones polynomial in knot theory, which leads to other new knot polynomials as well as connections between knot theory
Timeline_of_mathematics
ABEL POLYNOMIALS
ABEL POLYNOMIALS
Male
English
Anglicized form of Hebrew Abiy'el, ABIEL means "El (God) is (my) father." In the bible, this is the name of Saul's grandfather.
Female
English
Medieval short form of English Amabel, MABEL means "lovable."Â
Boy/Male
Hebrew
Exhalation of breath. The second son of Adam in the bible. The variant Able is used as an English...
Female
German
German form of Greek Barbara, BÄRBEL means "foreign; strange."
Male
English
 In the bible, this is the name of the second son of Adam and Eve who was killed by his jealous brother Cain. Anglicized form of Greek Habel, ABEL means "vanity," i.e. "transitory." Anglicized form of Hebrew Hebel, meaning "breath, breathing."
Male
English
Variant spelling of English Abel, ABELL means "vanity," i.e. "transitory."
Male
Hebrew
Variant spelling of Hebrew Abie, ABEY means "father of a multitude."
Biblical
a city; mourning,vanity; breath; transitoriness
Boy/Male
Indian
Healthy, Vanity, Breath, Breathing
Male
Hungarian
Hungarian form of Greek Habel, �BEL means "vanity," i.e. "transitory."
Male
Scandinavian
Scandinavian form of Hebrew Abiyshalowm, AXEL means "father of peace."Â
Biblical
mourning to the house of Maachah,meadow of the house of Maachah,also called ABEL-MAIM
Boy/Male
Hebrew
Exhalation of breath. The second son of Adam in the bible. The variant Able is used as an English...
Male
African
breath, vapor; transitoriness.
Male
Italian
Italian form of Hebrew Hebel, ABELE means "breath, breathing."
Surname or Lastname
English
English : variant spelling of Abel. Probably also an Americanized spelling of the same surname in other languages.
Boy/Male
American, Australian, British, Chinese, Christian, Danish, Dutch, English, Finnish, French, German, Hawaiian, Hebrew, Indian, Irish, Norwegian, Polish, Portuguese, Romanian, Swedish
Breath; Highborn and Steadfast; Child; Breathing Spirit; Son; Vapour
Male
English
Variant spelling of English Abel, ABLE means "vanity," i.e. "transitory."
Male
English
Breath
Boy/Male
Biblical American Hebrew
Vanity, breath, vapor. Also a city, mourning'.
ABEL POLYNOMIALS
ABEL POLYNOMIALS
Girl/Female
Indian
Valuable
Boy/Male
British, Indian, Romanian
Form of Arman or Harmanas
Boy/Male
Tamil
Vedima | வேதீமாஂ
Boy/Male
Indian
Servant of the expander, Extender
Girl/Female
Hindu, Indian
Removed Leaves on the Ground
Boy/Male
Anglo Saxon
Sharp.
Boy/Male
French Hebrew Italian
Surname or Lastname
English (Norfolk)
English (Norfolk) : variant spelling of Neve ‘nephew’.Scottish : from a place called Nevay in Angus.
Girl/Female
Muslim
Shining of gold
Surname or Lastname
English
English : of uncertain origin; Reaney derives it from an Old English personal name, Denebeald, an unrecorded compound of Dene-.
ABEL POLYNOMIALS
ABEL POLYNOMIALS
ABEL POLYNOMIALS
ABEL POLYNOMIALS
ABEL POLYNOMIALS
imp. & p. p.
of Abet
adv.
To childbed (in the phrase "brought abed," that is, delivered of a child).
imp. & p. p.
of Label
p. pr. & vb. n.
of Abet
v. t.
To affix a label to; to mark with a name, etc.; as, to label a bottle or a package.
p. pr. & vb. n.
of Label
superl.
Having sufficient power, strength, force, skill, means, or resources of any kind to accomplish the object; possessed of qualifications rendering competent for some end; competent; qualified; capable; as, an able workman, soldier, seaman, a man able to work; a mind able to reason; a person able to be generous; able to endure pain; able to play on a piano.
a.
Able to sway.
a.
Able to speak.
a.
To make able; to enable; to strengthen.
a.
Able to digest.
v. t.
To instigate or encourage by aid or countenance; -- used in a bad sense of persons and acts; as, to abet an ill-doer; to abet one in his wicked courses; to abet vice; to abet an insurrection.
superl.
Specially: Having intellectual qualifications, or strong mental powers; showing ability or skill; talented; clever; powerful; as, the ablest man in the senate; an able speech.
n.
A slip of silk, paper, parchment, etc., affixed to anything, usually by an inscription, the contents, ownership, destination, etc.; as, the label of a bottle or a package.
superl.
Legally qualified; possessed of legal competence; as, able to inherit or devise property.
v. t.
To affix in or on a label.