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Polynomial with indeterminate coefficients
generic polynomial refers usually to a polynomial whose coefficients are indeterminates. For example, if a, b, and c are indeterminates, the generic polynomial
Generic_polynomial
Topics referred to by the same term
point that contains all other points Generic polynomial, a polynomial whose coefficients are indeterminates Generic property, a formal definition of a property
Generic
Plane algebraic curve
mathematics, a polynomial lemniscate or polynomial level curve is a plane algebraic curve of degree 2n, constructed from a polynomial p with complex coefficients
Polynomial_lemniscate
Mathematical concept in polynomial theory
resultant of two polynomials is a polynomial expression of their coefficients that is equal to zero if and only if the polynomials have a common root
Resultant
Property holding for typical examples
statements, "A generic polynomial does not have a root at zero," or "A generic square matrix is invertible." As another example, a generic property of a
Generic_property
Pair of polynomial sequences
The Chebyshev polynomials are two sequences of orthogonal polynomials related to the cosine and sine functions, denoted as Tn(x) and Un(x). They can be
Chebyshev_polynomials
Unsolved problem in mathematics
century, is unsolved. There are some permutation groups for which generic polynomials are known, which define all algebraic extensions of ℚ having a particular
Inverse_Galois_problem
Equations of degree 5 or higher cannot be solved by radicals
resolvent. This is a univariate polynomial of degree six whose coefficients are polynomials in the coefficients of a generic quintic. A specific irreducible
Abel–Ruffini_theorem
Function of a polynomial's coefficients that gives information about its roots
a polynomial is an expression in its coefficients that provides information about its roots without explicitly calculating them. For a polynomial with
Discriminant
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
Number of times an object must be counted for making true a general formula
it appears in the multiset. For example, the number of times a given polynomial has a root at a given point is the multiplicity of that root. The notion
Multiplicity_(mathematics)
Greatest common divisor of polynomials
GCD or gcd) of two polynomials is a polynomial, of the highest possible degree, which is a factor of both the two original polynomials. This concept is
Polynomial greatest common divisor
Polynomial_greatest_common_divisor
is in GenP (generically polynomial time) if it never gives incorrect answers and if it gives correct answers in polynomial time on a generic set of inputs
Generic-case_complexity
Italian physicist and peace activist (1935–2026)
of any generic polynomial of arbitrary degree [F. Calogero, "Novel differential algorithm to evaluate all the zeros of any generic polynomial", J. Nonlinear
Francesco_Calogero
Formula for solving a quadratic equation
q are symmetric polynomials in α and β. Specifically, they are the elementary symmetric polynomials – any symmetric polynomial in α and β can
Quadratic_formula
Method of analyzing variables in software
gen-kill problems, or locally separable problems. Such problems have generic polynomial-time solutions. In addition to the reaching definitions and live variables
Data-flow_analysis
Arithmetic in a field with a finite number of elements
doi:10.2478/jee-2018-0059, S2CID 115440420 "Fast CRC Computation for Generic Polynomials Using PCLMULQDQ Instruction" (PDF). www.intel.com. 2009. Retrieved
Finite_field_arithmetic
Number of intersection points of algebraic curves and hypersurfaces
"complete", which in modern terminology would translate to generic. Since with generic polynomials, there are no points at infinity, and all multiplicities
Bézout's_theorem
(xy-yx)^{2}=-\det(xy-yx)I} for any 2-by-2-matrices x and y. Generic matrix ring Formanek, Edward (1991). The polynomial identities and invariants of n×n matrices. Regional
Central_polynomial
About polynomials in several variables
Jacobian conjecture is a conjecture concerning polynomials in several variables that states that if a polynomial function from an n-dimensional space to itself
Jacobian_conjecture
Error-correcting codes
polynomial c(x). The polynomial c(x) is constructed by multiplying the message polynomial m(x), which has degree k−1, with the generator polynomial g(x)
Reed–Solomon_error_correction
For a square matrix, the transpose of the cofactor matrix
invertible with inverse (det A)−1A. adj(A−1) = adj(A)−1. adj(A) is entrywise polynomial in A. In particular, over the real or complex numbers, the adjugate is
Adjugate_matrix
Species naming system
consisted of a generic name combined with a specific name that was from one to several words long. Together they formed a system of polynomial nomenclature
Binomial_nomenclature
Extension to the x86 instruction set
Archived from the original on 2019-08-06. "Fast CRC Computation for Generic Polynomials Using PCLMULQDQ" (PDF). Vlad Krasnov (2015-07-08). "Fighting Cancer:
CLMUL_instruction_set
On the number of common zeros of Laurent polynomials
Laurent polynomials, LA1,…,LAn. Consider a generic system of equations from these subspaces, that is: f1(x)=⋯=fn(x)=0, where each fi is a generic element
Bernstein–Kushnirenko_theorem
In algebra, a generic matrix ring is a sort of a universal matrix ring. We denote by F n {\displaystyle F_{n}} a generic matrix ring of size n with variables
Generic_matrix_ring
Canadian mathematician
Finite Fields. Cambridge University Press. ISBN 9780521498340. "Generic Polynomials: Constructive Aspects of the Inverse Galois Problem, by Christian
Noriko_Yui
{\displaystyle p(t)} . Results of this form are generically referred to as Positivstellensatz. Likewise, the Polynomial Matrix Spectral Factorization provides
Polynomial matrix spectral factorization
Polynomial_matrix_spectral_factorization
In mathematics, invariant of square matrices
determinant of a generic matrix is a polynomial in these indeterminates, with integer coefficients, called a generic determinant. More precisely, if M is
Determinant
Israeli mathematician (1921–1994)
of polynomial rings and related constructions. His results on algebras over uncountable set fields, ring of quotients and Morita equivalence, generic polynomial
Shimshon_Amitsur
Square matrices satisfy their characteristic equation
A2−5A−2I2=(7101522)−(5101520)−(2002)=(0000). For a generic 2×2 matrix, A=(abcd), the characteristic polynomial is given by p(λ) = λ2 − (a + d)λ + (ad − bc)
Cayley–Hamilton_theorem
1007/978-3-540-24849-1_14. Nicola Gambino; Joachim Kock (2013). "Polynomial functors and polynomial monads". Mathematical Proceedings of the Cambridge Philosophical
Container_(type_theory)
Concept in algebraic geometry
defined by some equations g1 = 0, ..., gr = 0, where each gi is in the polynomial ring k[x1,..., xn]. The affine scheme X is smooth of dimension m over
Smooth_scheme
Boolean satisfiability is NP-complete and therefore that NP-complete problems exist
NP-complete. That is, it is in NP, and any problem in NP can be reduced in polynomial time by a deterministic Turing machine to the Boolean satisfiability problem
Cook–Levin_theorem
Limiting case which is different from the rest of the class
root of a polynomial is sometimes said to be degenerate if it is a multiple root, since generically the n roots of an nth degree polynomial are all distinct
Degeneracy_(mathematics)
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
Branch of mathematics
geometrical problems. Classically, it studies zeros of multivariate polynomials; the modern approach generalizes this in a few different aspects. The
Algebraic_geometry
Generalization of algebraic variety
theory. The origins of algebraic geometry mostly lie in the study of polynomial equations over the real numbers. By the 19th century, it became clear
Scheme_(mathematics)
{\displaystyle \psi _{2m}\in 2y\mathbb {Z} [x,A,B]} . The division polynomials form a generic elliptic divisibility sequence over the ring Q [ x , y , A , B
Division_polynomials
Topology on prime ideals and algebraic varieties
non-linear irreducible polynomials. In this case, the spectrum consists of one closed point for each monic irreducible polynomial, and a generic point corresponding
Zariski_topology
Family of interpolation and clamping functions
upperlimit) return upperlimit; return x; } Starting with a generic third-order polynomial function and its first derivative: S 1 ( x ) = a 3 x 3 + a
Smoothstep
Networking protocol
Generic Stream Encapsulation, or GSE for short, is a data link layer protocol defined by DVB. GSE provides means to carry packet oriented protocols such
Generic_Stream_Encapsulation
Curve defined as zeros of polynomials
set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables
Algebraic_curve
Concept in algebraic geometry
geometry and computational geometry, general position is a notion of genericity for a set of points, or other geometric objects. It means the general
General_position
Partition of space by hyperplanes
−1.) This polynomial helps to solve some basic questions; see below. Another polynomial associated with A is the Whitney-number polynomial wA(x, y), defined
Arrangement_of_hyperplanes
Invariant of polynomial roots
defines the generic monic polynomial of degree n F(X)=Xn+∑i=1n(−1)iEiXn−i=∏i=1n(X−Xi), where Ei is the i th elementary symmetric polynomial. The symmetric
Resolvent_(Galois_theory)
Decision problem in computer science
is O(N2C). This solution does not count as polynomial time in complexity theory because B−A is not polynomial in the size of the problem, which is the number
Subset_sum_problem
Injective polynomial functions are bijective
Some partial converses to the Ax-Grothendieck Theorem: A generically surjective polynomial map of n {\displaystyle n} -dimensional affine space over
Ax–Grothendieck_theorem
Type of triangular sets of polynomial
respectively. Note that: (1) the content of the second polynomial is x2, which does not contribute to the generic points represented and thus can be removed; (2)
Regular_chain
Algorithmic complexity class
machine in exponential time, i.e., in O(2p(n)) time, where p(n) is a polynomial function of n. EXPTIME is one intuitive class in an exponential hierarchy
EXPTIME
Subfield of mathematical optimization
However, generic search algorithms are not guaranteed to find an optimal solution first, nor are they guaranteed to run quickly (in polynomial time). Since
Combinatorial_optimization
Artificial river barrier
A polynomial weir is a weir that has a geometry defined by a polynomial equation of any order n. In practice, most weirs are low-order polynomial weirs
Weir
Historical term in mathematics
calculus referred to the surprising similarity between seemingly unrelated polynomial equations and certain shadowy techniques used to prove them. These techniques
Umbral_calculus
Number of points needed to determine an algebraic curve
line goes through those two points. Likewise, a non-degenerate conic (polynomial equation in x and y with the sum of their powers in any term not exceeding
Cramer's theorem (algebraic curves)
Cramer's_theorem_(algebraic_curves)
Numerical analysis concept
wi are quadrature weights, and xi are the roots of the nth Legendre polynomial. This choice of quadrature weights wi and quadrature nodes xi is the unique
Gauss–Legendre_quadrature
Geometric group theory
reductions to power circuit division problems. There is a strongly generically polynomial time solution of the conjugacy problem for G. Andrew Brunner considered
Baumslag–Gersten_group
Result of commutative algebra
such that A {\displaystyle A} is a finitely generated module over the polynomial ring S = k [ y 1 , y 2 , … , y d ] {\displaystyle S=k[y_{1},y_{2},\ldots
Noether_normalization_lemma
Mathematical concept in algebraic geometry
rational functions on V. In classical algebraic geometry they are ratios of polynomials; in complex geometry these are meromorphic functions and their higher-dimensional
Function field of an algebraic variety
Function_field_of_an_algebraic_variety
Tool for analyzing divide-and-conquer algorithms
constant; the number of subproblems should be fixed T(n)=2T(n2)+nlogn non-polynomial difference between f(n) and nlogba (see below; extended version applies)
Master theorem (analysis of algorithms)
Master_theorem_(analysis_of_algorithms)
18 mathematical problems stated in 1998
Beltrán, Carlos; Pardo, Luis Miguel (2009). "Smale's 17th Problem: Average Polynomial Time to compute affine and projective solutions" (PDF). Journal of the
Smale's_problems
Fundamental operation on complex numbers
number: a2+b2 (or r2 in polar coordinates). If a root of a univariate polynomial with real coefficients is complex, then its complex conjugate is also
Complex_conjugate
Italian mathematician (1731–1807)
and X(400). Jensen, Christian U.; Ledet, Arne; Yui, Noriko (2002). Generic Polynomials: Constructive Aspects of the Inverse Galois Problem. Cambridge: Cambridge
Gian_Francesco_Malfatti
Branch of number theory
polynomial functions f we have the Newton series: f(x)=∑k=0∞(Δkf)(0)(xk), where (xk)=x(x−1)(x−2)⋯(x−k+1)k! is the kth binomial coefficient polynomial
P-adic_analysis
Solving an optimization problem with a quadratic objective function
problem in (weakly) polynomial time. This was explicitly shown in 1979 by Kozlov, Tarasov and Khachiyan. Ye and Tse present a polynomial-time algorithm, which
Quadratic_programming
logic and polynomial time". Theoretical Computer Science. 318 (1–2): 163–180. doi:10.1016/j.tcs.2003.10.018. Leivant, Daniel (2020). "A Generic Imperative
Implicit computational complexity
Implicit_computational_complexity
Field of asymmetric cryptographic primitives
Multivariate cryptography is the generic term for asymmetric cryptographic primitives based on multivariate polynomials over a finite field F {\displaystyle
Multivariate_cryptography
In algebra, module with a finite generating set
By the Hilbert–Serre theorem, there is a polynomial F such that PM(t)=F(t)∏(1−tdi)−1. Then F(1) is the generic rank of M. A finitely generated module over
Finitely_generated_module
Root-finding algorithm for polynomials
solving polynomial equations. In other words, the method can be used to solve numerically the equation f(x)=0, where f is a given polynomial, which can
Durand–Kerner_method
Code added to allow recovery of lost data
(Lagrange) polynomial p(x) of order k such that p(i) is equal to data symbol i. He then sends p(k), ..., p(n − 1). The receiver can now also use polynomial interpolation
Erasure_code
3d hypersurface of degree 5
a degree 5 {\displaystyle 5} homogeneous polynomial. One of the most studied examples is from the polynomial p ( x ) = x 0 5 + x 1 5 + x 2 5 + x 3 5 +
Quintic_threefold
Twenty-first letter in the Greek alphabet
φ(n) in number theory; also called Euler's phi function. The cyclotomic polynomial functions Φn(x) of algebra. The number of electrical phases in a power
Phi
Mathematical object studied in the field of algebraic geometry
an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. Modern definitions generalize
Algebraic_variety
Algebraic structure with addition and multiplication
complex numbers, but they may also be non-numerical objects such as polynomials, square matrices, functions, and power series. More formally, a ring
Ring_(mathematics)
Technique of representing an aggregate data structure
numbers label the finite types; in this way, the type constructors resemble polynomial functions. The derivative of a type constructor can therefore be formed
Zipper_(data_structure)
Topics referred to by the same term
used in Chinese martial arts and sport wushu Algebraic form (homogeneous polynomial), which generalises quadratic forms to degrees 3 and more, also known
Form
Partition of a graph by removing fewest possible edges
at T. When the weights are non-negative, the problem can be solved in polynomial time by the Stoer-Wagner algorithm. A generalization of the minimum cut
Minimum_cut
Quantum search algorithm
provide polynomial-time solutions for NP-complete problems (as the square root of an exponential function is still an exponential, not a polynomial function)
Grover's_algorithm
9th-century Arabic work on algebra
Al-Jabr provided an exhaustive account of solving for the positive roots of polynomial equations up to the second degree. It was the first text to teach elementary
Al-Jabr
Algebraic curve
y2+h(x)y=f(x) where f(x) is a polynomial of degree n = 2g + 1 > 4 or n = 2g + 2 > 4 with n distinct roots, and h(x) is a polynomial of degree < g + 2 (if the
Hyperelliptic_curve
States that the algebra of n by n matrices satisfies a certain identity of degree 2n
matrix rings are polynomial identity rings such that the smallest identity they satisfy has degree exactly 2n. The standard polynomial of degree n is S
Amitsur–Levitzki_theorem
prime. Considering the generic m × n matrix whose entries are algebraically independent variables x i,j, these minors are polynomials of degree r + 1. The
Determinantal_variety
Type of error-correcting code
evaluation of the polynomial on points in S {\displaystyle S} and interpolates that polynomial. Then it is simple to evaluate the polynomial at the point that
Locally_decodable_code
Computer hardware technology that uses quantum mechanics
as quantum parallelism. Peter Shor built on these results in 1994 with polynomial-time quantum algorithms for integer factorization and the discrete logarithm
Quantum_computing
Algebraic variety defined within an affine space
zeros over an algebraically closed field k of some family of polynomials in the polynomial ring k[x1,…,xn]. An affine variety is an affine algebraic set
Affine_variety
Software programming optimization technique
refraining redundant computations to accommodate any form of ambiguous CFG in polynomial time (Θ(n4) for left-recursive grammars and Θ(n3) for non left-recursive
Memoization
used as a proxy to establishing if a polynomial is convex. In contrast, deciding if a generic quartic polynomial of degree four (or higher even degree)
SOS-convexity
bracket ring is the subring of the ring of polynomials k[x11,...,xdn] generated by the d-by-d minors of a generic d-by-n matrix (xij). The bracket ring may
Bracket_ring
Matrices similar to diagonal matrices
annihilated by some polynomial (xn−λ1)⋯(xn−λk), which has no multiple root (since λj≠0) and is divided by the minimal polynomial of A. Over the complex
Diagonalizable_matrix
Gradient boosting machine learning library
(It arises from minimizing the second-order Taylor polynomial of L and completing the square. Minimizing this weighted squared error
XGBoost
Relation between genus, degree, and dimension of function spaces over surfaces
the Hilbert polynomial of line bundles on a curve. If a line bundle L {\displaystyle {\mathcal {L}}} is ample, then the Hilbert polynomial will give the
Riemann–Roch_theorem
C standard library header file
not part of the standard library. The header <tgmath.h> defines a type-generic macro for each mathematical function defined in <math.h> and <complex.h>
C_mathematical_functions
Characteristic class of oriented, real vector bundles
the fact that the Euler characteristic of the circle is 0. Vandermonde polynomial Thom isomorphism Generalized Gauss–Bonnet theorem Chern class Pontryagin
Euler_class
Indian-American mathematician
The Kirchhoff polynomial of G is a certain homogeneous polynomial whose degree is equal to the first betti number of G. These polynomials appear in the
Prakash_Belkale
Simple Lie group; the automorphism group of the octonions
octonion algebra. The variables must be non-commutative otherwise the second polynomial would be identically zero. Adding a representation of the 14 generators
G2_(mathematics)
Type of two-dimensional barcode
polynomial is x8+x4+x3+x2+1, corresponding to the decimal polynomial number 285 (1000111012), with initial root = 0 to obtain generator polynomials.
QR_code
Decomposition in multilinear algebra
form of the linear matrix pencil that the tensor represents. A simple polynomial-time algorithm exists for certifying that a tensor is of rank 1, namely
Tensor_rank_decomposition
Non-cryptographic hash function
constraints tend to have better dispersion properties. They improve the polynomial feedback characteristic when an FNV prime multiplies an intermediate hash
Fowler–Noll–Vo_hash_function
Branch of computational complexity theory
behaviour is called fixed parameter tractability, since, although both are polynomial time algorithms, the former is of the form O(n^c) independent of the parameter
Parameterized_complexity
Sum of inverse squares of natural numbers
elementary symmetric polynomials. Namely, we have a recurrence relation between the elementary symmetric polynomials and the power sum polynomials given as on
Basel_problem
Function used in signal processing
discrete-time windows. A kth-order B-spline basis function is a piece-wise polynomial function of degree k − 1 that is obtained by k-fold self-convolution of
Window_function
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GENERIC POLYNOMIAL
GENERIC POLYNOMIAL
GENERIC POLYNOMIAL
GENERIC POLYNOMIAL
GENERIC POLYNOMIAL
GENERIC POLYNOMIAL
GENERIC POLYNOMIAL
GENERIC POLYNOMIAL
GENERIC POLYNOMIAL
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