Journal of Pure and Applied Algebra 169 (2002) 229–248
www.elsevier.com/locate/jpaa
Eective equidimensional decomposition of ane varieties
Gabriela Jeronimo
∗;1
, Juan Sabia
1
Departamento de Matem
Ã
atica, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires,
Ciudad Universitaria, Pab. I, (1428), Buenos Aires, Argentina
Received 11 September 2000; received in revised form 28 February 2001
Communicated by M.-F. Roy
Abstract
In this paper we present a probabilistic algorithmwhich computes, froma ÿnite set of polyno-
mials deÿning an algebraic variety V , the decomposition of V into equidimensional components.
If V is deÿned by s polynomials in n variables of degrees bounded by an integer d ¿ n and
V =
r
‘=0
V
‘
is the equidimensional decomposition of V , the algorithmobtains in sequential time
bounded by s
O(1)
d
O(n)
, for each 0 6 ‘ 6 r, a set of n + 1 polynomials of degrees bounded by
deg (V
‘
) which deÿne V
‘
.
c
2002 Elsevier Science B.V. All rights reserved.
MSC: Primary 14Q20; secondary 68W30
0. Introduction
Dierent problems appearing nowadays are related to systems of polynomial equa-
tions. Some of these problems can be solved simply by deciding whether the associated
polynomial equation systemis consistent or not. However, when the systemis consis-
tent, it is sometimes necessary to describe the set of its solutions. The set of solutions
of a polynomial equation system is called an algebraic variety.
A well-known result states that any algebraic variety V over an algebraically closed
ÿeld K can be uniquely decomposed into a union of irreducible algebraic varieties
C
1
;:::;C
t
deÿnable by polynomials with coecients in K such that C
i
* C
j
for i = j.
∗
Corresponding author.
E-mail addresses: jeronimo@dm.uba.ar (G. Jeronimo), jsabia@dm.uba.ar (J. Sabia).
1
Partially supported by the following Argentinian research grants: CONICET: PIP ’97 4571; UBACyT:
EX TW80 (1998).
0022-4049/02/$ - see front matter
c
2002 Elsevier Science B.V. All rights reserved.
PII: S0022-4049(01)00083-4