zérus

Zérus is a term used in mathematical contexts to denote a zero or a root of a function or equation. In this sense, a zérus of a real-valued function f is a value x0 for which f(x0) = 0. When f is a polynomial, the zérus are called roots; over the complex numbers every polynomial of degree n has exactly n zeros counted with multiplicity, according to the Fundamental Theorem of Algebra. Zérus can be real or complex; real zeros correspond to x-intercepts of the graph, while nonreal zeros occur in conjugate pairs if the polynomial has real coefficients.

Multiplicity describes how many times a given zérus appears as a solution. If a polynomial P(x) has a root x0 of multiplicity m, then (x − x0)^m divides P(x) but (x − x0)^(m+1) does not. In several variables, the zero set Z(f) = {x | f(x) = 0} is called a zero set or an algebraic variety in algebraic geometry, and its structure encodes geometric and algebraic information about f.

In numerical analysis, finding zérus typically means locating approximations to roots when closed-form solutions are unavailable. Common methods include bisection, Newton-Raphson, and the secant method, each with conditions guaranteeing convergence.

Examples include the zeros of sine, x = kπ, and the zeros of x^2 − 1, namely x = ±1. The concept of zérus is foundational across mathematics, underpinning equation solving, analysis, and geometry.