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In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, a zero (also sometimes called a root) of a real-, complex-, or generally
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. The input of a vector-valued function could ...
f, is a member x of the domain of f such that f(x) ''vanishes'' at x; that is, the function f attains the value of 0 at x, or equivalently, x is a solution to the equation f(x) = 0. A "zero" of a function is thus an input value that produces an output of 0. A root of a
polynomial In mathematics, a polynomial is a Expression (mathematics), mathematical expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addit ...
is a zero of the corresponding polynomial function. The fundamental theorem of algebra shows that any non-zero
polynomial In mathematics, a polynomial is a Expression (mathematics), mathematical expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addit ...
has a number of roots at most equal to its degree, and that the number of roots and the degree are equal when one considers the complex roots (or more generally, the roots in an algebraically closed extension) counted with their multiplicities. For example, the polynomial f of degree two, defined by f(x)=x^2-5x+6=(x-2)(x-3) has the two roots (or zeros) that are 2 and 3. f(2)=2^2-5\times 2+6= 0\textf(3)=3^2-5\times 3+6=0. If the function maps real numbers to real numbers, then its zeros are the x-coordinates of the points where its graph meets the ''x''-axis. An alternative name for such a point (x,0) in this context is an x-intercept.


Solution of an equation

Every
equation In mathematics, an equation is a mathematical formula that expresses the equality of two expressions, by connecting them with the equals sign . The word ''equation'' and its cognates in other languages may have subtly different meanings; for ...
in the
unknown Unknown or The Unknown may refer to: Film and television Film * The Unknown (1915 comedy film), ''The Unknown'' (1915 comedy film), Australian silent film * The Unknown (1915 drama film), ''The Unknown'' (1915 drama film), American silent drama ...
x may be rewritten as :f(x)=0 by regrouping all the terms in the left-hand side. It follows that the solutions of such an equation are exactly the zeros of the function f. In other words, a "zero of a function" is precisely a "solution of the equation obtained by equating the function to 0", and the study of zeros of functions is exactly the same as the study of solutions of equations.


Polynomial roots

Every real polynomial of odd degree has an odd number of real roots (counting multiplicities); likewise, a real polynomial of even degree must have an even number of real roots. Consequently, real odd polynomials must have at least one real root (because the smallest odd whole number is 1), whereas even polynomials may have none. This principle can be proven by reference to the intermediate value theorem: since polynomial functions are continuous, the function value must cross zero, in the process of changing from negative to positive or vice versa (which always happens for odd functions).


Fundamental theorem of algebra

The fundamental theorem of algebra states that every polynomial of degree n has n complex roots, counted with their multiplicities. The non-real roots of polynomials with real coefficients come in conjugate pairs. Vieta's formulas relate the coefficients of a polynomial to sums and products of its roots.


Computing roots

There are many methods for computing accurate approximations of roots of functions, the best being
Newton's method In numerical analysis, the Newton–Raphson method, also known simply as Newton's method, named after Isaac Newton and Joseph Raphson, is a root-finding algorithm which produces successively better approximations to the roots (or zeroes) of a ...
, see Root-finding algorithm. For
polynomial In mathematics, a polynomial is a Expression (mathematics), mathematical expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addit ...
s, there are specialized algorithms that are more efficient and may provide all roots or all real roots; see Polynomial root-finding and Real-root isolation. Some polynomial, including all those of degree no greater than 4, can have all their roots expressed algebraically in terms of their coefficients; see Solution in radicals.


Zero set

In various areas of mathematics, the zero set of a function is the set of all its zeros. More precisely, if f:X\to\mathbb is a real-valued function (or, more generally, a function taking values in some additive group), its zero set is f^(0), the inverse image of \ in X. Under the same hypothesis on the codomain of the function, a level set of a function f is the zero set of the function f-c for some c in the codomain of f. The zero set of a
linear map In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that p ...
is also known as its kernel. The cozero set of the function f:X\to\mathbb is the complement of the zero set of f (i.e., the subset of X on which f is nonzero).


Applications

In
algebraic geometry Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ...
, the first definition of an
algebraic variety Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the solution set, set of solutions of a system of polynomial equations over the real number, ...
is through zero sets. Specifically, an affine algebraic set is the
intersection In mathematics, the intersection of two or more objects is another object consisting of everything that is contained in all of the objects simultaneously. For example, in Euclidean geometry, when two lines in a plane are not parallel, their ...
of the zero sets of several polynomials, in a
polynomial ring In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring formed from the set of polynomials in one or more indeterminates (traditionally also called variables) with coefficients in another ring, ...
k\left _1,\ldots,x_n\right/math> over a field. In this context, a zero set is sometimes called a ''zero locus''. In
analysis Analysis (: analyses) is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle (38 ...
and
geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
, any closed subset of \mathbb^n is the zero set of a smooth function defined on all of \mathbb^n. This extends to any
smooth manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One may ...
as a corollary of paracompactness. In
differential geometry Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of Calculus, single variable calculus, vector calculus, lin ...
, zero sets are frequently used to define
manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a N ...
s. An important special case is the case that f is a smooth function from \mathbb^p to \mathbb^n. If zero is a regular value of f, then the zero set of f is a smooth manifold of dimension m=p-n by the regular value theorem. For example, the unit m-
sphere A sphere (from Ancient Greek, Greek , ) is a surface (mathematics), surface analogous to the circle, a curve. In solid geometry, a sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...
in \mathbb^ is the zero set of the real-valued function f(x)=\Vert x \Vert^2-1.


See also

* Root-finding algorithm * Bolzano's theorem, a continuous function that takes opposite signs at the end points of an interval has at least a zero in the interval. * Gauss–Lucas theorem, the complex zeros of the derivative of a polynomial lie inside the convex hull of the roots of the polynomial. * Marden's theorem, a refinement of Gauss–Lucas theorem for polynomials of degree three * Sendov's conjecture, a conjectured refinement of Gauss-Lucas theorem * zero at infinity * Zero crossing, property of the graph of a function near a zero * Zeros and poles of holomorphic functions


References


Further reading

* {{MathWorld , title=Root , urlname=Root Elementary mathematics Functions and mappings 0 (number)