@MISC{ALGGEO_1.MIZ,
  AUTHOR = {Watase, Yasushige},
  TITLE = {{I}ntroduction to {A}lgebraic {G}eometry},
  SECTION1 = {Preliminaries},
  SECTION2 = {Algebraic Sets},
  SECTION3 = {The Ideal of a Set of Points},
  SECTION4 = {Irreducible Components of an Algebraic Set},
  DAY = {30},
  MONTH = {June},
  YEAR = {2023},
  ADDRESS1 = {Suginami-ku Matsunoki 6, 3-21 Tokyo\\Japan},
  SUMMARY = {A classical algebraic geometry is study of zero points of system of multivariate       polynomials and those zero points would be corresponding to points, lines, curves, surfaces 
  in an affine space. \par
  In this article we give some basic definition of the area of affine algebraic geometry such as    algebraic set, ideal of set of points, and those properties
  according to \cite{Fulton69} in the Mizar system \cite{Mizar-State-2015}.\par
  Affine space used in the geometry normally written as $k^n$ is not formalized and 
  one need to identify a point of the space as n-tuple of $k$. 
  A formalization of $n$-tuple is in terms of a function$:\ n \longrightarrow k$, 
  where $k$ is the same as the scalar ring or field of polynomials which are defining a     geometrical object and $n = \{0,1,\dots, n-1\}$. 
  The same manner occurres while evaluate multivariate polynomial by 
  $n$-tuple of numbers \cite{POLYNOM2.ABS}. \par
  This formalization aims at providing basic notions of the field which enable to formalize 
  geometric object such as algebraic curves which is used in for instance coding theory 
  \cite{Stichtenoth08} as well as further formalization of the fields in the Mizar system.},
MSC2010 = {14-01,14H50,68V20},
KEYWORDS = {Affine Algebraic Set; multivariate Polynomial}}


@BOOK{Fulton69,
 AUTHOR={Fulton, Williama},
 TITLE={Algebraic Curves. An Introduction to Algebraic Geometry},
 PUBLISHER={The Benjamin/Cummings Publishing Company},
 YEAR=1969}

@BOOK{Stichtenoth08},
 AUTHOR={Stichtenoth, Henning},
 TITLE={Algebraic Function Fields and Codes},
 PUBLISHER={Springer},
 YEAR=2008}}
