zero set

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In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no general consensus abo ...
, a zero (also sometimes called a root) of a
real Real may refer to: * Reality Reality is the sum or aggregate of all that is real or existent within a system, as opposed to that which is only Object of the mind, imaginary. The term is also used to refer to the ontological status of things, ind ...
-,
complex The UCL Faculty of Mathematical and Physical Sciences is one of the 11 constituent faculties of University College London , mottoeng = Let all come who by merit deserve the most reward , established = , type = Public university, Public rese ...

-, or generally
vector-valued function A vector-valued function, also referred to as a vector function, is a function (mathematics), mathematical function of one or more variables whose range of a function, range is a set of multidimensional Euclidean vector, vectors or infinite-dimensi ...

$f$, 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 *Doma ...
of $f$ such that $f\left(x\right)$ ''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) Image:SaltInWaterSolutionLiquid.jpg, upMaking a saline water solution by dissolving Salt, table salt (sodium chloride, NaCl) in water. The salt is the solute and the water the solvent. In chemistry ...
to the equation $f\left(x\right) = 0$. A "zero" of a function is thus an input value that produces an output of 0. A root of a
polynomial In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

is a zero of the corresponding
polynomial function In mathematics, a polynomial is an expression (mathematics), expression consisting of variable (mathematics), variables (also called indeterminate (variable), indeterminates) and coefficients, that involves only the operations of addition, subtra ...
. The
fundamental theorem of algebra The fundamental theorem of algebra states that every non- constant single-variable polynomial In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, s ...
shows that any non-zero
polynomial In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

has a number of roots at most equal to its
degree Degree may refer to: As a unit of measurement * Degree symbol (°), a notation used in science, engineering, and mathematics * Degree (angle), a unit of angle measurement * Degree (temperature), any of various units of temperature measurement ...
, 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 (mathematics), field is algebraically closed if every Degree of a polynomial, non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a Zero of a function, root in . Examples As an example, t ...
) counted with their
multiplicities In mathematics, the multiplicity of a member of a multiset is the number of times it appears in the multiset. For example, the number of times a given polynomial has a Root of a function, root at a given point is the multiplicity of that root. The ...
. For example, the polynomial $f$ of degree two, defined by $f(x)=x^2-5x+6$ has the two roots $2$ and $3$, since $f(2) = 2^2 - 5 \cdot 2 + 6 = 0\quad\textrm\quad f(3) = 3^2 - 5 \cdot 3 + 6 = 0.$ If the function maps real numbers to real numbers, then its zeros are the $x$-coordinates of the points where its
graph Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties *Graph (topology), a topological space resembling a graph in the sense of discret ...

meets the ''x''-axis. An alternative name for such a point $\left(x,0\right)$ in this context is an $x$-intercept.

# Solution of an equation

Every
equation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

in the
unknown Unknown or The Unknown may refer to: Film * The Unknown (1915 comedy film), ''The Unknown'' (1915 comedy film), a silent boxing film * The Unknown (1915 drama film), ''The Unknown'' (1915 drama film) * The Unknown (1927 film), ''The Unknown'' (19 ...
$x$ may be rewritten as :$f\left(x\right)=0$ 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 Degree may refer to: As a unit of measurement * Degree symbol (°), a notation used in science, engineering, and mathematics * Degree (angle), a unit of angle measurement * Degree (temperature), any of various units of temperature measurement ...
has an odd number of real roots (counting
multiplicities In mathematics, the multiplicity of a member of a multiset is the number of times it appears in the multiset. For example, the number of times a given polynomial has a Root of a function, root at a given point is the multiplicity of that root. The ...
); 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 function, 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 complex conjugate, 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 function In mathematics, a polynomial is an expression (mathematics), expression consisting of variable (mathematics), variables (also called indeterminate (variable), indeterminates) and coefficients, that involves only the operations of addition, subtra ...
s, frequently requires the use of specialised or approximation techniques (e.g., Newton's method). However, some polynomial functions, including all those of degree of a polynomial, degree no greater than 4, can have all their roots expressed algebraic function, algebraically in terms of their coefficients (for more, see algebraic solution).

# Zero set

In various areas of mathematics, the zero set of a function (mathematics), 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 Abelian group, additive group), its zero set is $f^\left(0\right)$, 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 topology, topological properties. For example, a level set of a function $f$ is the zero set of $f-c$. The cozero set of $f$ is the complement (set theory), complement of the zero set of $f$ (i.e., the subset of $X$ on which $f$ is nonzero).

## Applications

In algebraic geometry, the first definition of an algebraic variety is through zero sets. Specifically, an affine algebraic set is the set intersection, intersection of the zero sets of several polynomials, in a polynomial ring $k\left\left[x_1,\ldots,x_n\right\right]$ over a field (mathematics), field. In this context, a zero set is sometimes called a ''zero locus''. In Mathematical analysis, analysis and geometry, any closed set, 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 as a corollary of paracompactness. In differential geometry, zero sets are frequently used to define manifolds. 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 Submersion_(mathematics)#Local_normal_form, regular value theorem. For example, the unit $m$-sphere in $\mathbb^$ is the zero set of the real-valued function $f\left(x\right)=\Vert x \Vert^2-1$.