Zero of a function

A zero of a function $f$ is an argument $a$ for which $f(a)=0$.

The zero set of a function $f$ is the preimage $f^{-1}(\{0\})$, consisting of all its zeros of a function. For a real- or complex-valued continuous function, it is a closed set, because the preimage of the closed singleton $\{0\}$ under a continuous function is closed. A nonzero holomorphic function on a connected open set has isolated zeros by the identity theorem for holomorphic functions.

A root of a polynomial $f$ is a scalar $a$ satisfying $f(a)=0$ in the chosen field or ring. For $p(X)=(X+1)(X-2{)}^2$ over the real numbers, the distinct roots of a polynomial are $-1$ and $2$. The second has multiplicity of a root two, so there are three roots when counted with multiplicity.

The domain matters. $X^2+1$ has no real polynomial root, but has the two complex number roots $i,-i$. The fundamental theorem of algebra guarantees a polynomial root in the complex numbers for every nonconstant complex polynomial, and complete factorization counts its degree of a polynomial many roots with multiplicity of a root. It does not promise that those roots are real or that they have a simple expression.

A polynomial root that is an integer. For a polynomial with integer coefficients and nonzero constant term, any integer root divides that constant term, by the rational root theorem.

Two real polynomials with only real roots of a polynomial interlace when their ordered roots alternate, counting multiplicity of a root. For equal polynomial degrees, one ordering is the displayed chain; the opposite ordering is also permitted. Consecutive degrees give the analogous chain with one extra root at each end for the higher-degree polynomial. Strict interlacing uses strict inequalities. Weak interlacing permits common roots and follows as a limit of strict interlacing.

For a monic polynomial $p(x)={\prod }_i(x-a_i)$ with distinct ordered real roots of a polynomial, each critical point $\xi$ lies between two consecutive roots by Rolle's theorem. Its logarithmic derivative gives ${\sum }_i(\xi -a_i{)}^{-1}=0$. Implicit differentiation gives the displayed formula, so moving any root to the right moves every ordered critical point to the right, while the roots remain distinct. Iterating this statement for derivatives preserves componentwise ordering of their roots. Weak inequalities extend by continuity to repeated roots.

Differentiation preserves the ordering of interlacing roots of polynomials of equal or consecutive polynomial degrees. Iteration gives the result for every derivative for which both polynomials are nonconstant; common roots are allowed by a limiting argument. This is a result about the locations of roots of a polynomial, distinct from the Markov inequality for polynomial derivatives. Theorem A of Dimitar K. Dimitrov's account states the first-derivative result.

A root of a polynomial lying on the complex unit circle. For a numerical amplification polynomial, a simple such root gives an undamped oscillatory mode. A repeated one supplies polynomial growth in the step index and fails the root condition for a multistep method.

A positive root of a real polynomial $p$ is a real number $r>0$ for which $p(r)=0$. Zero is excluded. A count of distinct positive roots counts each such number once; a count with multiplicity of a root counts a root of order $m$ exactly $m$ times. Multiplication by a power of $x$ does not change either positive-root count.

For a polynomial $f$ over a field, $f(a)=0$ exactly when $X-a$ divides $f(X)$. Polynomial division by the monic linear factor givesThe constant polynomial remainder is thus the value obtained by evaluation. For $p(X)=X^3-3X^2+4$, the root of a polynomial $2$ gives factor $X-2$ and polynomial quotient $X^2-X-2$. The latter also vanishes at $2$, explaining the repeated factor. This provides a practical way to move between evaluation and factorization.

Multiplicity records how many times an object occurs when repetitions carry mathematical information.

For a nonzero polynomial over a field, the multiplicity of a root $a$ is the largest integer $m\ge 1$ such that $(X-a{)}^m$ divides it. Write $p(X)=(X-a{)}^{m}q(X)$ with $q(a)\ne 0$. For real polynomials, continuity keeps the sign of $q$ fixed sufficiently near $a$, so an odd integer $m$ makes the graph cross the axis and an even integer $m$ makes it touch without crossing.

The right-hand panel above replaces $(X-2{)}^2$ by $(X-2{)}^2-\epsilon$. For $\epsilon =0.04$, its two roots of a polynomial are $1.8$ and $2.2$; for $\epsilon =-0.04$, they are $2\pm 0.2i$. The unperturbed double root lies between these two cases. Small coefficient changes can therefore split a repeated root into distinct real roots or move it off the real axis, illustrating why repeated-root problems can be numerically sensitive.

A root of a polynomial is double when its multiplicity of a root is exactly two. Equivalently, $p(x)=(x-x_0{)}^{2}q(x)$ with $q(x_0)\ne 0$. A generic small perturbation splits it according to square-root splitting of a double polynomial root.

A root is multiple when its multiplicity of a root is greater than one.

A common root of polynomials $f$ and $g$ is a scalar $a$ satisfying $f(a)=g(a)=0$.

A root of a polynomial lying in the complex numbers. It may be real: a nonreal complex root has nonzero imaginary part.