Set of zeros

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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 .


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 .


  • 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.


  • 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 .


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

  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