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 would suggest. For example, x 10 + 3 x 3 + 1 {\displaystyle x^{10}+3x^{3}+1} is a sparse polynomial as it is a trinomial with a degree of 10 {\displaystyle 10} .
10-478: [REDACTED] Look up binomial in Wiktionary, the free dictionary. Binomial may refer to: In mathematics [ edit ] Binomial (polynomial) , a polynomial with two terms Binomial coefficient , numbers appearing in the expansions of powers of binomials Binomial QMF , a perfect-reconstruction orthogonal wavelet decomposition Binomial theorem ,
20-434: A univariate binomial) can be written in the form where a and b are numbers , and m and n are distinct non-negative integers and x is a symbol which is called an indeterminate or, for historical reasons, a variable . In the context of Laurent polynomials , a Laurent binomial , often simply called a binomial , is similarly defined, but the exponents m and n may be negative. More generally,
30-404: A binomial may be written as: Sparse polynomial The motivation for studying sparse polynomials is to concentrate on the structure of a polynomial's monomials instead of its degree, as one can see, for instance, by comparing Bernstein-Kushnirenko theorem with Bezout's theorem . Research on sparse polynomials has also included work on algorithms whose running time grows as a function of
40-462: A sequence of two or more words or phrases in the same grammatical category, having some semantic relationship and joined by some syntactic device In biology [ edit ] Binomial nomenclature , a Latin two-term name for a species, such as Sequoia sempervirens In finance [ edit ] Binomial options pricing model , a numerical method for the valuation of options In politics [ edit ] Binomial voting system ,
50-793: A simple structure, which is also reflected in the structure of the solutions of certain related differential equations . Additionally, a sparse positivstellensatz exists for univariate sparse polynomials. It states that the non-negativity of a polynomial can be certified by sos polynomials whose degree only depends on the number of monomials of the polynomial. Sparse polynomials oftentimes come up in sum or difference of powers equations. The sum of two cubes states that ( x + y ) ( x 2 − x y + y 2 ) = x 3 + y 3 {\displaystyle (x+y)(x^{2}-xy+y^{2})=x^{3}+y^{3}} . Here x 3 + y 3 {\displaystyle x^{3}+y^{3}}
60-461: A theorem about powers of binomials Binomial type , a property of sequences of polynomials Binomial series , a mathematical series In probability and statistics [ edit ] Binomial distribution , a type of probability distribution Binomial process Binomial test , a test of significance In computing science [ edit ] Binomial heap , a data structure In linguistics [ edit ] Binomial pair ,
70-596: A voting system used in the parliamentary elections of Chile between 1989 and 2013 See also [ edit ] List of factorial and binomial topics Topics referred to by the same term [REDACTED] This disambiguation page lists articles associated with the title Binomial . If an internal link led you here, you may wish to change the link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=Binomial&oldid=1237750173 " Category : Disambiguation pages Hidden categories: Short description
80-876: Is a sparse polynomial since out of the 16 {\displaystyle 16} possible terms, only 2 {\displaystyle 2} appear. Other examples include the identities ( x − y ) ∑ k = 0 N − 1 x k y N − 1 − k = x N − y N {\displaystyle (x-y)\sum _{k=0}^{N-1}x^{k}y^{N-1-k}=x^{N}-y^{N}} and also ( x + y ) ∑ k = 0 2 n ( − 1 ) k x k y 2 n − k = x 2 n + 1 + y 2 n + 1 {\displaystyle (x+y)\sum _{k=0}^{2n}(-1)^{k}x^{k}y^{2n-k}=x^{2n+1}+y^{2n+1}} where
90-427: Is different from Wikidata All article disambiguation pages All disambiguation pages Binomial (polynomial) In algebra , 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. A binomial is a polynomial which is the sum of two monomials. A binomial in a single indeterminate (also known as
100-541: The number of terms rather than on the degree, for problems including polynomial multiplication , division , root-finding algorithms , and polynomial greatest common divisors . Sparse polynomials have also been used in pure mathematics, especially in the study of Galois groups , because it has been easier to determine the Galois groups of certain families of sparse polynomials than it is for other polynomials. The algebraic varieties determined by sparse polynomials have
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