Set of zeros

A set of zeros is a subset of the domain of a function and contains all the arguments that are mapped to zero. Sets of zeros can be found in many areas of mathematics. The determination of the set of zeros of a function is both part of school mathematics and part of the Riemann Hypothesis and thus one of the Millennium problems .

definition

Given a function with a domain and a target set , with a specially marked zero element . Then the amount is called ${\displaystyle f\colon D\to Z}$ ${\displaystyle D}$ ${\displaystyle Z}$ ${\displaystyle 0\in Z}$

${\displaystyle N=\{x\in D\mid f(x)=0\}}$

the set of zeros of the function . ${\displaystyle f}$

Remarks

• The set of zeros contains all zeros of the function and is therefore exactly the set of levels of the function to the value . ${\displaystyle 0}$
• Because of this, the set of zeros is a value of the original image function that belongs to it . Because their argument is one element here, it is about the fiber from over . ${\displaystyle N=f^{-1}(\{0\})}$ ${\displaystyle f}$ ${\displaystyle f}$ ${\displaystyle \{0\}}$ ${\displaystyle N}$ ${\displaystyle f}$ ${\displaystyle 0}$
• The target set must have at least the structure of a magma with one , i.e. a set with a two-digit link and a neutral element . Examples of such structures are groups , rings , solids and vector spaces . In most cases, the target set will be the real or complex numbers. ${\displaystyle 0}$
• In a group homomorphism with a (additively written) group , the set of zeros of is also called the kernel of . This also applies in particular to algebraic structures that expand such groups, such as rings or vector spaces as target sets. ${\displaystyle f\colon G\to H}$ ${\displaystyle H}$ ${\displaystyle f}$ ${\displaystyle \ker f}$ ${\displaystyle f}$

Examples

• The polynomial function with ${\displaystyle f\colon \mathbb {R} \to \mathbb {R} }$

${\displaystyle f(x)=x^{2}-1}$

owns the set of zeros . ${\displaystyle N=\{-1,1\}}$

• The sine function with ${\displaystyle f\colon \mathbb {R} \to \mathbb {R} }$

${\displaystyle f(x)=\sin(x)}$

owns the set of zeros . ${\displaystyle N=\{\pi k\,|\,k\in \mathbb {Z} \}}$

• The function with ${\displaystyle f\colon \mathbb {R} ^{2}\to \mathbb {R} }$

${\displaystyle f(x,y)=x^{2}+y^{2}-1}$

has the unit circle as the set of zeros .

Varieties

If a body , the polynomial ring in n variables is over and is a subset, then in algebraic geometry one considers the set of zeros of : ${\displaystyle K}$ ${\displaystyle K[X_{1},\ldots ,X_{n}]}$ ${\displaystyle K}$ ${\displaystyle I\subset K[X_{1},\ldots ,X_{n}]}$ ${\displaystyle I}$

${\displaystyle {\mathfrak {V}}(I):=\{(x_{1},\ldots ,x_{n})\in K^{n}|\,f(x_{1},\ldots ,x_{n})=0{\text{ für alle }}f\in I\}}$

This is called the variety of . This is the average of the zero sets of all polynomial functions from polynomials . ${\displaystyle I}$ ${\displaystyle K^{n}\rightarrow K}$ ${\displaystyle I}$

Z sets

If a topological space , then a subset is called a Z-set if it is the set of zeros of a continuous function , that is, if it holds for a continuous function . The Z in Z set comes from the English word zero for zero. Since there is a closed set and since archetypes of closed sets are closed again under continuous mappings, all Z-sets must be closed. ${\displaystyle X}$ ${\displaystyle Y\subset X}$ ${\displaystyle f\colon X\rightarrow \mathbb {R} }$ ${\displaystyle Y=\{x\in X\mid f(x)=0\}}$ ${\displaystyle f}$ ${\displaystyle \{0\}\subset \mathbb {R} }$

Individual evidence

1. ↑ Ernst Kunz : Introduction to Commutative Algebra and Algebraic Geometry. Vieweg (1980), ISBN 3-528-07246-6 , Chapter I, Definition 1.7.
2. ^ Johann Cigler , Hans-Christian Reichel : Topology. A basic lecture. Bibliographisches Institut, Mannheim et al. 1978, ISBN 3-411-00121-6 ( BI university pocket books 121), § 4.6.

Web links

Wiktionary: set of zeros  - explanations of meanings, word origins, synonyms, translations

• Zero Set in Mathworld