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
the set of zeros of the function .
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 .
- 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 .
- 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.
- 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.
Examples
- The polynomial function with
- owns the set of zeros .
- The sine function with
- owns the set of zeros .
- The function with
- 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 :
This is called the variety of . This is the average of the zero sets of all polynomial functions from polynomials .
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.
Individual evidence
- ↑ Ernst Kunz : Introduction to Commutative Algebra and Algebraic Geometry. Vieweg (1980), ISBN 3-528-07246-6 , Chapter I, Definition 1.7.
- ^ 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.