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In mathematics, a zero (also sometimes called a root) of a
real Real may refer to: Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish colonial real Music Albums * ''Real'' (L'Arc-en-Ciel album) (2000) * ''Real'' (Bright album) (2010) ...
-,
complex Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...
-, or generally vector-valued function f, is a member x of the
domain Domain may refer to: Mathematics *Domain of a function, the set of input values for which the (total) function is defined ** Domain of definition of a partial function ** Natural domain of a partial function **Domain of holomorphy of a function * ...
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 the
solution Solution may refer to: * Solution (chemistry), a mixture where one substance is dissolved in another * Solution (equation), in mathematics ** Numerical solution, in numerical analysis, approximate solutions within specified error bounds * Solutio ...
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 is a zero of the corresponding polynomial function. The fundamental theorem of algebra shows that any non-zero polynomial 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 In mathematics, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a root in . Examples As an example, the field of real numbers is not algebraically closed, because ...
) counted with their multiplicities. For example, the polynomial f of degree two, defined by f(x)=x^2-5x+6 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 the
unknown Unknown or The Unknown may refer to: Film * ''The Unknown'' (1915 comedy film), a silent boxing film * ''The Unknown'' (1915 drama film) * ''The Unknown'' (1927 film), a silent horror film starring Lon Chaney * ''The Unknown'' (1936 film), a ...
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 Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous g ...
, 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

Computing roots of functions, for example polynomial functions, frequently requires the use of specialised or
approximation An approximation is anything that is intentionally similar but not exactly equal to something else. Etymology and usage The word ''approximation'' is derived from Latin ''approximatus'', from ''proximus'' meaning ''very near'' and the prefix '' ...
techniques (e.g., Newton's method). However, some polynomial functions, including all those of degree no greater than 4, can have all their roots expressed algebraically in terms of their coefficients (for more, see
algebraic solution A solution in radicals or algebraic solution is a closed-form expression, and more specifically a closed-form algebraic expression, that is the solution of a polynomial equation, and relies only on addition, subtraction, multiplication, divisio ...
).


Zero set

In various areas of mathematics, the zero set of a
function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriente ...
is the set of all its zeros. More precisely, if f:X\to\mathbb is a
real-valued function In mathematics, a real-valued function is a function whose values are real numbers. In other words, it is a function that assigns a real number to each member of its domain. Real-valued functions of a real variable (commonly called ''real fu ...
(or, more generally, a function taking values in some
additive group An additive group is a group of which the group operation is to be thought of as ''addition'' in some sense. It is usually abelian, and typically written using the symbol + for its binary operation. This terminology is widely used with structure ...
), its zero set is f^(0), the inverse image of \ in X. The term ''zero set'' is generally used when there are infinitely many zeros, and they have some non-trivial
topological properties In topology and related areas of mathematics, a topological property or topological invariant is a property of a topological space that is invariant under homeomorphisms. Alternatively, a topological property is a proper class of topological space ...
. For example, a level set of a function f is the zero set of f-c. The cozero set of f is the
complement A complement is something that completes something else. Complement may refer specifically to: The arts * Complement (music), an interval that, when added to another, spans an octave ** Aggregate complementation, the separation of pitch-class ...
of the zero set of f (i.e., the subset of X on which f is nonzero). The zero set of a linear map is also called
kernel Kernel may refer to: Computing * Kernel (operating system), the central component of most operating systems * Kernel (image processing), a matrix used for image convolution * Compute kernel, in GPGPU programming * Kernel method, in machine learni ...
.


Applications

In
algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrica ...
, 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 set of solutions of a system of polynomial equations over the real or complex numbers. ...
is through zero sets. Specifically, an
affine algebraic set Affine may describe any of various topics concerned with connections or affinities. It may refer to: * Affine, a relative by marriage in law and anthropology * Affine cipher, a special case of the more general substitution cipher * Affine com ...
is the intersection of the zero sets of several polynomials, in a polynomial ring k\left _1,\ldots,x_n\right/math> over a
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...
. 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 (3 ...
and geometry, any
closed subset In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points. In a complete metric space, a ...
of \mathbb^n is the zero set of a smooth function defined on all of \mathbb^n. This extends to any smooth manifold as a corollary of paracompactness. In differential geometry, 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 ...
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 In mathematics, a submersion is a differentiable map between differentiable manifolds whose differential is everywhere surjective. This is a basic concept in differential topology. The notion of a submersion is dual to the notion of an immersion ...
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 in \mathbb^ is the zero set of the real-valued function f(x)=\Vert x \Vert^2-1.


See also

* Marden's theorem * Root-finding algorithm *
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. It is named after Blagovest Sendov. Th ...
* Vanish at infinity * Zero crossing *
Zeros and poles In complex analysis (a branch of mathematics), a pole is a certain type of singularity of a complex-valued function of a complex variable. In some sense, it is the simplest type of singularity. Technically, a point is a pole of a function if i ...


References


Further reading

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