In mathematics, a sparse polynomial (also lacunary polynomial or fewnomial) is a

polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exa ...

that has far fewer terms than its

degree Degree may refer to: As a unit of measurement * Degree (angle), a unit of angle measurement ** Degree of geographical latitude ** Degree of geographical longitude * Degree symbol (°), a notation used in science, engineering, and mathematics ...

and number of variables would suggest. Examples include *

monomial 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 called power product, is a product of powers of variables with nonnegative integer exponent ...

s, polynomials with only one term, * binomials, polynomials with only two terms, and *

trinomial In elementary algebra, a trinomial is a polynomial consisting of three terms or monomials. Examples of trinomial expressions # 3x + 5y + 8z with x, y, z variables # 3t + 9s^2 + 3y^3 with t, s, y variables # 3ts + 9t + 5s with t, s variables # a ...

s, polynomials with only three terms. Research on sparse polynomials has included work on

algorithm In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific Computational problem, problems or to perform a computation. Algorithms are used as specificat ...

s whose running time grows as a function of the number of terms rather than on the degree, for problems including

polynomial multiplication In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An example ...

root-finding algorithms In mathematics and computing, a root-finding algorithm is an algorithm for finding zeros, also called "roots", of continuous functions. A zero of a function , from the real numbers to real numbers or from the complex numbers to the complex numbers ...

, and

polynomial greatest common divisor In algebra, the greatest common divisor (frequently abbreviated as GCD) of two polynomials is a polynomial, of the highest possible degree, that is a factor of both the two original polynomials. This concept is analogous to the greatest common di ...

s. Sparse polynomials have also been used in pure mathematics, especially in the study of

Galois group In mathematics, in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension. The study of field extensions and their relationship to the pol ...

s, 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 Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. Mo ...

determined by sparse polynomials have a simple structure, which is also reflected in the structure of the solutions of certain related

differential equation In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...

References

Polynomials {{algebra-stub