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In mathematics, a sparse polynomial (also lacunary polynomial or fewnomial) is a polynomial that has far fewer terms than its degree and number of variables
Sparse_polynomial
In mathematics, a polynomial with two terms
a binomial is a polynomial that is the sum of two terms, each of which is a monomial. It is the simplest kind of a sparse polynomial after the monomials
Binomial_(polynomial)
the CRC of the message modulo a sparse polynomial which is a multiple of the CRC polynomial. For CRC-32, the polynomial x123 + x111 + x92 + x84 + x64 +
Computation of cyclic redundancy checks
Computation_of_cyclic_redundancy_checks
Polynomial with only one term
In mathematics, a monomial is, roughly speaking, a polynomial which has only one term. Two definitions of a monomial may be encountered: A monomial, also
Monomial
Problem of determining whether polynomials are identical
runtime. A sparse PIT has at most m {\displaystyle m} nonzero monomial terms. A sparse PIT can be deterministically solved in polynomial time of the
Polynomial_identity_testing
Class of pseudorandom number generators
efficient implementation in software without the excessive use of sparse polynomials. They generate the next number in their sequence by repeatedly taking
Xorshift
Topics referred to by the same term
measure of the extent that a pattern contains gaps Lacunary polynomial, or sparse polynomial Petrovsky lacuna, in mathematics Laguna (disambiguation) This
Lacuna
Sparse binary polynomial hashing (SBPH) is a generalization of Bayesian spam filtering that can match mutating phrases as well as single words. SBPH is
Sparse binary polynomial hashing
Sparse_binary_polynomial_hashing
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
Tool used in probabilistic polynomial identity testing
probabilistic polynomial identity testing. Identity testing is the problem of determining whether a given multivariate polynomial is the 0-polynomial, the polynomial
Schwartz–Zippel_lemma
Class of problems solvable in polynomial time
exists a sparse language that is P-complete, then L = P. P is contained in BQP; it is unknown whether this containment is strict. Polynomial-time algorithms
P_(complexity)
Problem of sorting pairs of numbers by their sum
of the problem include transit fare minimisation, VLSI design, and sparse polynomial multiplication. As with comparison sorting and integer sorting more
X_+_Y_sorting
languages in P/poly are sparse, there is a polynomial-time Turing reduction from any language in P/poly to a sparse language. There is a Turing reduction (as
Sparse_language
Discrete Fourier transform algorithm
computes such transformations by factorizing the DFT matrix into a product of sparse (mostly zero) factors. As a result, it manages to reduce the complexity
Fast_Fourier_transform
Mathematical construct in computer algebra
of polynomial equations because FGML does not take into account the sparsity of involved matrices. This has been fixed by the introduction of sparse FGLM
Gröbner_basis
Algebraic encoding of graph connectivity
The Tutte polynomial, also called the dichromate or the Tutte–Whitney polynomial, is a graph polynomial. It is a polynomial in two variables which plays
Tutte_polynomial
Function in algebraic graph theory
The chromatic polynomial is a graph polynomial studied in algebraic graph theory, a branch of mathematics. It counts the number of graph colorings as a
Chromatic_polynomial
Graph with almost the max amount of edges
4)-sparse. Streinu and Theran show that testing (k,l)-sparsity may be performed in polynomial time when k and l are integers and 0 ≤ l < 2k. For a graph
Dense_graph
Polynomial that has three terms
Monomial Binomial Multinomial Simple expression Compound expression Sparse polynomial Quadratic expressions are not always trinomials, the expressions'
Trinomial
Statistical analysis technique
Sparse principal component analysis (SPCA or sparse PCA) is a technique used in statistical analysis and, in particular, in the analysis of multivariate
Sparse_PCA
Finding the roots of polynomials is a long-standing problem that has been extensively studied throughout the history and substantially influenced the
Polynomial_root-finding
Numerical method for solving physical or engineering problems
defined with polynomial and even non-polynomial shapes (e.g., ellipse or circle). Examples of methods that use higher degree piecewise polynomial basis functions
Finite_element_method
Concepts from linear algebra
the roots of a polynomial with degree 5 or more. (Generality matters because any polynomial with degree n is the characteristic polynomial of some companion
Eigenvalues_and_eigenvectors
Theorem in computational complexity theory
there exists a sparse language, such that a polynomial-time algorithm exists to solve the SAT problem by making O(1) queries to the sparse language oracle
Mahaney's_theorem
Data-driven algorithm
Sparse identification of nonlinear dynamics (SINDy) is a data-driven algorithm for obtaining dynamical systems from data. Given a series of snapshots of
Sparse identification of non-linear dynamics
Sparse_identification_of_non-linear_dynamics
Computational problem in graph theory
pseudo-polynomial and weakly polynomial is that a pseudo-polynomial bound may be polynomial in U {\displaystyle U} , but for a weakly polynomial bound
Maximum_flow_problem
Problem in combinatorial optimization
pseudo-polynomial time algorithm using dynamic programming. There is a fully polynomial-time approximation scheme, which uses the pseudo-polynomial time
Knapsack_problem
Moving average and polynomial regression method for smoothing data
regression or local polynomial regression, also known as moving regression, is a generalization of the moving average and polynomial regression. Its most
Local_regression
Signal processing technique
Compressed sensing (also known as compressive sensing, compressive sampling, or sparse sampling) is a signal processing technique for efficiently acquiring and
Compressed_sensing
Methodic assignment of colors to elements of a graph
Birkhoff introduced the chromatic polynomial to study the coloring problem, which was generalised to the Tutte polynomial by W. T. Tutte, both of which are
Graph_coloring
Factorization algorithm
number field sieve is super-polynomial but sub-exponential in the size of the input. Suppose f is a k-degree polynomial over Q {\textstyle \mathbb {Q}
General_number_field_sieve
Set of problems solved by small circuits
languages in P/poly are sparse languages, there is a polynomial-time Turing reduction from any language in P/poly to a sparse language. Adleman's theorem
P/poly
Relation between algebraic varieties and polynomial ideals
conditions for the existence of solutions to systems of multivariate polynomial equations over an algebraically closed field (such as the complex numbers
Hilbert's_Nullstellensatz
Family of graphs whose shallow minors are sparse graphs
minors are sparse graphs. Many natural families of sparse graphs have bounded expansion. A closely related but stronger property, polynomial expansion
Bounded_expansion
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
Unrelated vertices in graphs
approximated to within any approximation ratio c < 1 in polynomial time; similar polynomial-time approximation schemes exist in any family of graphs
Independent set (graph theory)
Independent_set_(graph_theory)
Complexity class
problem with only polynomial overhead. If P is different from co-NP, then all of the co-NP-complete problems are not solvable in polynomial time. If there
Co-NP-complete
Task of computing complete subgraphs
the maximum as can be found in polynomial time. Although much of this work has focused on independent sets in sparse graphs, a case that does not make
Clique_problem
Decision problem in computer science
This solution does not count as polynomial time in complexity theory because B − A {\displaystyle B-A} is not polynomial in the size of the problem, which
Subset_sum_problem
Mapping a graph onto itself without changing edge-vertex connectivity
list of generators, is polynomial-time equivalent to the graph isomorphism problem, and therefore solvable in quasi-polynomial time, that is with running
Graph_automorphism
Mathematical result
random. If you keep rolling the dice, you will eventually obtain one in polynomial random time. The proof below is based on the course notes of Afonso Bandeira
Johnson–Lindenstrauss_lemma
Multiparty cryptographic process
verifiable secret sharing protocol to share the results of two random polynomial functions. Every party then verifies all the shares they received. If
Distributed_key_generation
Partition of a graph's nodes into 2 disjoint subsets
minimum cut that separates the source and the sink are equal. There are polynomial-time methods to solve the min-cut problem, notably the Edmonds–Karp algorithm
Cut_(graph_theory)
Unsolved problem in structural complexity theory
Turing reductions, the existence of a sparse NP-complete language would imply an unexpected collapse of the polynomial hierarchy. As evidence towards the
Berman–Hartmanis_conjecture
Number divisible only by 1 and itself
and the AKS primality test, which always produces the correct answer in polynomial time but is too slow to be practical. Particularly fast methods are available
Prime_number
delta-squared process — most useful for linearly converging sequences Minimum polynomial extrapolation — for vector sequences Richardson extrapolation Shanks transformation
List of numerical analysis topics
List_of_numerical_analysis_topics
including linear (standard and generalized) and nonlinear (quadratic, polynomial and general), as well as the SVD. Recent versions also include support
SLEPc
Function in discrete mathematics
converting between sample values and the coefficients of a trigonometric polynomial that interpolates those values. It is therefore a basic tool for numerical
Discrete_Fourier_transform
Game in algorithmic game theory
be done in polynomial time, and for a graph with a bounded treewidth, this is also true for finding an optimal correlated equilibrium. Sparse games are
Succinct_game
Array of numbers
the eigenvalues of a square matrix are the roots of its characteristic polynomial, det ( λ I − A ) {\displaystyle \det(\lambda I-A)} . Matrix theory is
Matrix_(mathematics)
Function defined by multiple sub-functions
function composed of power-law sub-functions Spline, a function composed of polynomial sub-functions, often constrained to be smooth at the joints between pieces
Piecewise_function
Algorithms for computing Gröbner bases
Jean-Charles Faugère, computes the Gröbner basis of an ideal of a multivariate polynomial ring. The algorithm uses the same mathematical principles as the Buchberger
Faugère's F4 and F5 algorithms
Faugère's_F4_and_F5_algorithms
class, a non-uniform analogue of the class NP of problems solvable in polynomial time by a non-deterministic Turing machine. It is the non-deterministic
NP/poly
Cycles in a graph that generate all cycles
positive weights, the minimum weight cycle basis may be constructed in polynomial time. In planar graphs, the set of bounded cycles of an embedding of the
Cycle_basis
the discrete logarithm problem Polynomial long division: an algorithm for dividing a polynomial by another polynomial of the same or lower degree Risch
List_of_algorithms
Discrete analog of a derivative
the polynomial is 36x. Subtracting out the third term: Without any pairwise differences, it is found that the 4th and final term of the polynomial is the
Finite_difference
Mathematical concept
positive integers n such that for all integers k with 1 ≤ k < n, the polynomial k2 − k + n produces a prime number. When k is equal to n, the value cannot
Lucky_numbers_of_Euler
Statement in complex analysis
represented as a product involving its zeroes and an exponential of a polynomial. It is named for Jacques Hadamard. The theorem may be viewed as an extension
Hadamard factorization theorem
Hadamard_factorization_theorem
Branch of mathematics
graphs, and especially the chromatic polynomial, the Tutte polynomial and knot invariants. The chromatic polynomial of a graph, for example, counts the
Algebraic_graph_theory
Sparse graph with strong connectivity
In graph theory, an expander graph is a sparse graph that has strong connectivity properties, quantified using vertex, edge or spectral expansion. Expander
Expander_graph
Subfield of mathematical optimization
over convex sets). Many classes of convex optimization problems admit polynomial-time algorithms, whereas mathematical optimization is in general NP-hard
Convex_optimization
Computer algebra system
finite field elements, multivariable polynomials, rational functions, or polynomials modulo other polynomials. The main areas of application are multivariate
Fermat (computer algebra system)
Fermat_(computer_algebra_system)
Computer system for solving algebra problems
fundamental integer and polynomial operations, such as the Schönhage–Strassen algorithm for fast multiplication of integers and polynomials. Integer factorization
Magma (computer algebra system)
Magma_(computer_algebra_system)
Partition of a graph's nodes into cliques
number in perfect graphs in polynomial time. Another class of graphs in which the minimum clique cover can be found in polynomial time are the triangle-free
Clique_cover
Method to solve optimization problems
polynomial-time algorithm? Does LP admit a strongly polynomial-time algorithm to find a strictly complementary solution? Does LP admit a polynomial-time
Linear_programming
Unsolved problem in the mathematics of graph coloring
when the diameter is bounded by a polynomial function of n, this suggests that the mixing time might also be polynomial. In his 2007 doctoral dissertation
Cereceda's_conjecture
Concept in computational complexity theory
as verifiers. A language L is in NEXPTIME if and only if there exist polynomials p and q, and a deterministic Turing machine M, such that For all x and
NEXPTIME
Software for approximating the roots of a polynomial with arbitrarily high precision
use of multiprecision. "Mpsolve takes advantage of sparsity, and has special hooks for polynomials that can be evaluated efficiently by straight-line
MPSolve
Infinite integer series where the next number is the sum of the two preceding it
as Fibonacci polynomials are derived from the Fibonacci numbers, the Lucas polynomials L n ( x ) {\displaystyle L_{n}(x)} are a polynomial sequence derived
Lucas_number
Subset of artificial intelligence
polynomial time. There are two kinds of time complexity results: Positive results show that a certain class of functions can be learned in polynomial
Machine_learning
Vector satisfying some of the criteria of an eigenvector
{\displaystyle A} must be in F {\displaystyle F} . That is, the characteristic polynomial f ( x ) {\displaystyle f(x)} must factor completely into linear factors;
Generalized_eigenvector
German mathematician
Zbl 0944.65131 Gatermann, Karin; Huber, Birkett (2002), "A family of sparse polynomial systems arising in chemical reaction systems", Journal of Symbolic
Karin_Gatermann
Square matrix used to represent a graph or network
and A2 are similar and therefore have the same minimal polynomial, characteristic polynomial, eigenvalues, determinant and trace. These can therefore
Adjacency_matrix
On collapse of the polynomial hierarchy if NP is in non-uniform polynomial time class
Schuler, If NP has Polynomial-Size Circuits, then MA = AM Kannan, R. (1982). "Circuit-size lower bounds and non-reducibility to sparse sets". Information
Karp–Lipton_theorem
Area of discrete mathematics
chromatic polynomial is a polynomial that counts the number of graph colorings as a function of the number of colors. The Tutte polynomial is a two-variable
Graph_theory
Optimization problem
exchange for a sparser x, basis pursuit denoising is preferred. Basis pursuit problems can be converted to linear programming problems in polynomial time and
Basis_pursuit
Topics referred to by the same term
Routh–Hurwitz matrix, a square matrix constructed with coefficients of a real polynomial Parity-check matrix is often called H-matrix. This disambiguation page
H-matrix
Maximum number of colors obtainable by a greedy graph coloring algorithm
given graph is at least k, for a fixed constant k, can be performed in polynomial time, by searching for all possible k-atoms that might be subgraphs of
Grundy_number
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
UP: Unambiguous Polynomial-Time. Hemaspaandra, Lane A.; Rothe, Jörg (June 1997). "Unambiguous Computation: Boolean Hierarchies and Sparse Turing-Complete
UP_(complexity)
Indian mathematician (born 1956)
an ISI highly cited researcher. He invented one of the first provably polynomial time algorithms for linear programming, which is generally referred to
Narendra_Karmarkar
Formal language in computational complexity theory
This may be proven by considering a polynomial time algorithm for 3-SAT. This result can be extended to sparse languages. If L is a unary language, then
Unary_language
(having both directed and undirected edges). The program is solvable in polynomial time if the graph has all undirected or all directed edges. Variants include
List_of_NP-complete_problems
Mapping arbitrary data to fixed-size values
division by a polynomial modulo 2 instead of an integer to map n bits to m bits. In this approach, M = 2m, and we postulate an mth-degree polynomial Z(x) = xm
Hash_function
Algorithm to be run on quantum computers
integer factorization problem in polynomial time, whereas the best known classical algorithms take super-polynomial time. It is unknown whether these
Quantum_algorithm
Set of numbers used in the smoothsort algorithm
{5}}\right)/2} are the roots of the quadratic polynomial x 2 − x − 1 = 0 {\displaystyle x^{2}-x-1=0} . The Leonardo polynomials L n ( x ) {\displaystyle L_{n}(x)}
Leonardo_number
Mathematical optimization problem restricted to integers
a fixed constant, then the feasibility problem can be solved in time polynomial in m and log V. This is trivial for the case n=1. The case n=2 was solved
Integer_programming
Logical formulation of graph properties
the n {\displaystyle n} -vertex graphs that model the property, with polynomial delay (as a function of n {\displaystyle n} ) per graph. A similar analysis
Logic_of_graphs
Numerical integration method
roots of a Chebyshev polynomial and these values are used to construct a polynomial approximation for the function. This polynomial is then integrated exactly
Clenshaw–Curtis_quadrature
Property in graph theory
The biclique-free graph families form one of the most general types of sparse graph family. They arise in incidence problems in discrete geometry, and
Biclique-free_graph
Routines for performing common linear algebra operations
chronological order of definition and publication, as well as the degree of the polynomial in the complexities of algorithms; Level 1 BLAS operations typically take
Basic Linear Algebra Subprograms
Basic_Linear_Algebra_Subprograms
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
Australian and American mathematician (born 1975)
locations of the roots and critical points of a complex polynomial, in the special case of polynomials with sufficiently high degree. In 2024 and 2025, Tao
Terence_Tao
Infinite prime numbers of the form a^2+b^4
Heath-Brown and Xiannan Li in 2017. In particular, they proved that the polynomial a 2 + b 4 {\displaystyle a^{2}+b^{4}} represents infinitely many primes
Friedlander–Iwaniec_theorem
Property of artificial neural networks
hidden layer. It states that if the layer's activation function is non-polynomial (which is true for common choices like the sigmoid function or ReLU),
Universal approximation theorem
Universal_approximation_theorem
Economical computational problem
polymatrix games, approximating a Nash equilibrium with polynomial precision is PPAD-hard, even for sparse win-lose games. Rank-1 bimatrix games have payoff
Nash_equilibrium_computation
Computational complexity of quantum algorithms
as the set of problems solvable by a (deterministic) Turing machine in polynomial time. Similarly, quantum complexity classes may be defined using quantum
Quantum_complexity_theory
Integers occurring in the coefficients of the Taylor series of 1/cosh t
function. The Euler numbers are related to a special value of the Euler polynomials, namely E n = 2 n E n ( 1 2 ) . {\displaystyle E_{n}=2^{n}E_{n}({\tfrac
Euler_numbers
Quantum algorithm for solving systems of linear equations
factoring algorithm and Grover's search algorithm. Assuming the system is sparse, has a low condition number κ {\displaystyle \kappa } , and that the user
HHL_algorithm
Numerical eigenvalue calculation
p(A)v_{1}} for some polynomial p {\displaystyle p} of degree at most m − 1 {\displaystyle m-1} ; the coefficients of that polynomial are simply the coefficients
Lanczos_algorithm
SPARSE POLYNOMIAL
SPARSE POLYNOMIAL
Female
English
English variant form of French Cerise, SHARISE means "cherry."Â
Boy/Male
Afghan, Arabic, Iranian, Muslim, Parsi
Pious; Pure; Chaste; Holy
Surname or Lastname
Irish (Kerry)
Irish (Kerry) : Anglicized form of Gaelic Mac Saoghair, which in turn may be a patronymic from a Gaelicized form of the Old English personal name Saeger (see 2 below).English : patronymic from a Middle English personal name Saher or Seir (see Sayer 1).Americanized form of French Cyr.Richard Sears came to Plymouth, MA, from England about 1630.
Boy/Male
Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Telugu
Feel; Healthy; Touch
Surname or Lastname
English
English : variant of Sparks.
Girl/Female
Hindu, Indian
Touch
Male
English
Short form of English unisex Paisley, PAISE means "church."Â
Surname or Lastname
Portuguese
Portuguese : occupational name from soeiro ‘swineherd’, Latin suerius.English : patronymic from a nickname for someone with reddish hair, from Anglo-Norman French sor ‘chestnut (color)’.
Boy/Male
American, British, English
Gallant
Surname or Lastname
English
English : metonymic occupational name for someone who made bags or purses or for an official in charge of expenditure, from Middle English purse (via Old English from Latin bursa).Scottish : variant of Purser.
Surname or Lastname
English
English : variant of Spear.
Surname or Lastname
English
English : variant of Speake.
Surname or Lastname
English
English : patronymic from Spark 1.
Surname or Lastname
English
English : patronymic from Spire 1.
Surname or Lastname
English
English : nickname for a frugal person, from Middle English spare ‘sparing’, ‘frugal’.
Surname or Lastname
English
English : patronymic from Spear.
Boy/Male
Anglo Saxon Welsh
Spares.
Surname or Lastname
English (Suffolk)
English (Suffolk) : unexplained.
Surname or Lastname
English
English : variant spelling of Pass.French : possibly a nickname from passe ‘sparrow’.
Surname or Lastname
English
English : from the Norman personal name Serlo, Germanic Sarilo, Serilo. This was probably originally a byname cognate with Old Norse Sorli, and akin to Old English searu ‘armor’, meaning perhaps ‘defender’, ‘protector’.
SPARSE POLYNOMIAL
SPARSE POLYNOMIAL
Boy/Male
Muslim
The guided one
Girl/Female
Hindu, Indian
Hindu
Boy/Male
Indian, Sanskrit, Tamil
Without Any Faults
Boy/Male
American, Australian
He Enlightens
Boy/Male
Hindu
Light, Bright
Girl/Female
American, British, English
Darling; Dearly Loved
Boy/Male
Indian
Cool, Sweet, Intelligent
Male
Hebrew
(זֶבַח) Variant spelling of Hebrew Zebach, ZEVACH means "a slaying."
Girl/Female
Tamil
Girl/Female
American, Australian, Chinese, Christian, Greek, Welsh
Pearl; Based on the Abbreviation Meg
SPARSE POLYNOMIAL
SPARSE POLYNOMIAL
SPARSE POLYNOMIAL
SPARSE POLYNOMIAL
SPARSE POLYNOMIAL
n.
One who parses.
adv.
In a scattered or sparse manner.
n.
To emit sparks; to throw off ignited or incandescent particles; to shine as if throwing off sparks; to emit flashes of light; to scintillate; to twinkle; as, the blazing wood sparkles; the stars sparkle.
superl.
Not refined; rough; rude; unpolished; gross; indelicate; as, coarse manners; coarse language.
n.
Brilliancy; luster; as, the sparkle of a diamond.
v. t.
To inclose in a hearse; to entomb.
n.
The right of bowling again at a full set of pins, after having knocked all the pins down in less than three bowls. If all the pins are knocked down in one bowl it is a double spare; in two bowls, a single spare.
v. t.
Held in reserve, to be used in an emergency; as, a spare anchor; a spare bed or room.
v. t.
To emit in the form or likeness of sparks.
imp. & p. p.
of Parse
n.
A fine sieve; a searce.
imp. & p. p.
of Spare
v. t.
Being over and above what is necessary, or what must be used or reserved; not wanted, or not used; superfluous; as, I have no spare time.
adv.
Sparsely; scatteredly; here and there.
v. t.
To sprinkle; to moisten by sprinkling; as, to sparge paper.
v. t.
Scanty; not abundant or plentiful; as, a spare diet.
superl.
Large in bulk, or composed of large parts or particles; of inferior quality or appearance; not fine in material or close in texture; gross; thick; rough; -- opposed to fine; as, coarse sand; coarse thread; coarse cloth; coarse bread.
v. t.
To sift through a sarse.
n.
One who spares.
superl.
Thinly scattered; set or planted here and there; not being dense or close together; as, a sparse population.