# 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}$

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

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.