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# Zero of a function

In mathematics, a zero, also sometimes called a root, of a real-, complex- or generally vector-valued function f is a member x of the domain of f such that f(x) vanishes at x; that is, x is a solution of the equation f(x). [1]

## Abelian group

In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written.

## Affine variety

In algebraic geometry, an affine variety over an algebraically closed field k is the zero-locus in the affine ''n''-space k^n of some finite family of polynomials of n variables with coefficients in k that generate a prime ideal.

## Algebraic function

In mathematics, an algebraic function is a function that can be defined as the root of a polynomial equation.

## Algebraic geometry

Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials.

## Algebraic solution

An algebraic solution or solution in radicals is a closed-form expression, and more specifically a closed-form algebraic expression, that is the solution of an algebraic equation in terms of the coefficients, relying only on addition, subtraction, multiplication, division, raising to integer powers, and the extraction of nth roots (square roots, cube roots, and other integer roots).

## Algebraic variety

Algebraic varieties are the central objects of study in algebraic geometry.

## Algebraically closed field

In abstract algebra, an algebraically closed field F contains a root for every non-constant polynomial in F, the ring of polynomials in the variable x with coefficients in F.

## Approximation

An approximation is anything that is similar but not exactly equal to something else.

## Cartesian coordinate system

A Cartesian coordinate system is a coordinate system that specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular directed lines, measured in the same unit of length.

## Closed set

In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set.

## Complement (set theory)

In set theory, the complement of a set refers to elements not in.

## Complex conjugate

In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign.

## Complex number

A complex number is a number that can be expressed in the form, where and are real numbers, and is a solution of the equation.

## Continuous function

In mathematics, a continuous function is a function for which sufficiently small changes in the input result in arbitrarily small changes in the output.

## Degree of a polynomial

The degree of a polynomial is the highest degree of its monomials (individual terms) with non-zero coefficients.

## Differentiable manifold

In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a linear space to allow one to do calculus.

## Differential geometry

Differential geometry is a mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra to study problems in geometry.

## Domain of a function

In mathematics, and more specifically in naive set theory, the domain of definition (or simply the domain) of a function is the set of "input" or argument values for which the function is defined.

## Equation

In mathematics, an equation is a statement of an equality containing one or more variables.

## Equation solving

In mathematics, to solve an equation is to find its solutions, which are the values (numbers, functions, sets, etc.) that fulfill the condition stated by the equation, consisting generally of two expressions related by an equality sign.

## Field (mathematics)

In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined, and behave as when they are applied to rational and real numbers.

## Function (mathematics)

In mathematics, a function was originally the idealization of how a varying quantity depends on another quantity.

## Fundamental theorem of algebra

The fundamental theorem of algebra states that every non-constant single-variable polynomial with complex coefficients has at least one complex root.

## Graph of a function

In mathematics, the graph of a function f is, formally, the set of all ordered pairs, and, in practice, the graphical representation of this set.

## Image (mathematics)

In mathematics, an image is the subset of a function's codomain which is the output of the function from a subset of its domain.

## Intermediate value theorem

In mathematical analysis, the intermediate value theorem states that if a continuous function, f, with an interval,, as its domain, takes values f(a) and f(b) at each end of the interval, then it also takes any value between f(a) and f(b) at some point within the interval.

## Intersection (set theory)

In mathematics, the intersection A ∩ B of two sets A and B is the set that contains all elements of A that also belong to B (or equivalently, all elements of B that also belong to A), but no other elements.

## Level set

In mathematics, a level set of a real-valued function ''f'' of ''n'' real variables is a set of the form that is, a set where the function takes on a given constant value c. When the number of variables is two, a level set is generically a curve, called a level curve, contour line, or isoline.

## Manifold

In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point.

## Marden's theorem

In mathematics, Marden's theorem, named after Morris Marden but proven much earlier by Jörg Siebeck, gives a geometric relationship between the zeroes of a third-degree polynomial with complex coefficients and the zeroes of its derivative.

## Mathematics

Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.

## Multiplicity (mathematics)

In mathematics, the multiplicity of a member of a multiset is the number of times it appears in the multiset.

## Newton's method

In numerical analysis, Newton's method (also known as the Newton–Raphson method), named after Isaac Newton and Joseph Raphson, is a method for finding successively better approximations to the roots (or zeroes) of a real-valued function.

## Paracompact space

In mathematics, a paracompact space is a topological space in which every open cover has an open refinement that is locally finite.

## Polynomial

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

## Polynomial ring

In mathematics, especially in the field of abstract algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variables) with coefficients in another ring, often a field.

## Prentice Hall

Prentice Hall is a major educational publisher owned by Pearson plc.

## Real number

In mathematics, a real number is a value of a continuous quantity that can represent a distance along a line.

## Real-valued function

In mathematics, a real-valued function is a function whose values are real numbers.

## Sendov's conjecture

In mathematics, Sendov's conjecture, sometimes also called Ilieff's conjecture, concerns the relationship between the locations of roots and critical points of a polynomial function of a complex variable.

## Smoothness

In mathematical analysis, the smoothness of a function is a property measured by the number of derivatives it has that are continuous.

## Sphere

A sphere (from Greek σφαῖρα — sphaira, "globe, ball") is a perfectly round geometrical object in three-dimensional space that is the surface of a completely round ball (viz., analogous to the circular objects in two dimensions, where a "circle" circumscribes its "disk").

## Submersion (mathematics)

In mathematics, a submersion is a differentiable map between differentiable manifolds whose differential is everywhere surjective.

## Topology

In mathematics, topology (from the Greek τόπος, place, and λόγος, study) is concerned with the properties of space that are preserved under continuous deformations, such as stretching, crumpling and bending, but not tearing or gluing.

## Vanish at infinity

In mathematics, a function on a normed vector space is said to vanish at infinity if For example, the function defined on the real line vanishes at infinity.

## Vector-valued function

A vector-valued function, also referred to as a vector function, is a mathematical function of one or more variables whose range is a set of multidimensional vectors or infinite-dimensional vectors.

## Vieta's formulas

In mathematics, Vieta's formulas are formulas that relate the coefficients of a polynomial to sums and products of its roots.

## Zero crossing

A zero-crossing is a point where the sign of a mathematical function changes (e.g. from positive to negative), represented by a intercept of the axis (zero value) in the graph of the function.

## Zeros and poles

In mathematics, a zero of a function is a value such that.

## References

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