From: "Ariel Waissbein" <thewatas@ciudad.com.ar>
Subject: Re: proof
Date: Tue, 13 Jul 1999 14:24:53 -0300
Newsgroups: sci.math.symbolic
Keywords: Effective solutions of systems of polynomial equations (literature)

As the thread shows it is not necessary to have as many
equations as unknowns to have a solution, think of linear
equations and you'll have some more examples where
$n$ equations in $n$ unknowns do not have a solution,
or they have an infinite number or only one; hence it neither
is sufficient. Now, you might also know of Galois' result
which shows that there exists no solution by radicals for an
equation defined by a single univariate polynomial.

If what you are asking is for conditions to have a unique
solution, or a solution at all you also need to specify
the ground field. On that aspect over algebraicaly closed
fields there are efficient versions of Hilbert's Nullstellensatz
(e.g. {Brownawell87}, {Caniglia-Galligo-Heintz88}, {Kollar88}
and the new {KrickPardo96} many of these papers
can be found on the net). (Hilbert's Nullstellensatz states
conditions for the set of zeros  of (a finite number of) polynomial
equations being empty).
Algorithms solving the Nullstellensatz problem are unefficiently
implemented in MAPLE, MAGMA and other general purpose
softwares relying on rewriting preocesses (e.g. Gr\"obner
bases or resultants); a new generation of algorithms
(see http://hilbert.matesco.unican.es/tera ) have been proved to
be efficient in the sense of ``good geometric conditioning=fast
solution").

Try finding any text on Algebraic Geometry to grasp the basic
notions you need to understand the problem: have in mind the
dimension of the varieties
$V_k=V(F_1,...,F_k)=V_{k-1}\cap\{F_k^{-1}(0)\}$ defined
by the polynomials $F_1,...,F_s\in k[X_1,...,X_n]$.
There are many ``good'' properties you can look for to attain
some valuable results in the direction you point out such as
equidimensional varieties, regular sequences, Complete
intesection varieties, the Cohen--Macaulay condition, ...
Also take a peep into bertini's theorems which prove that
roughly speaking you can almost restrict yourself to the case
of $n$ equations in $n$ unknowns.


I hope to have cleared out your problems or at least pointed
out some directions, if you have any questions, please feel
free to ask.

Ariel.

(some) Basic Biblio
Cox, Little, and O'Shea  ``ideals Varieties and algorithms"
UTM, SpringerVerlag.


Kunz ``Intro to Commutative Algebra and algebraic geometry"
Bifkhauser.


I. Shaffarevich ``Algebraic Geometry"
Springer.

PAPERS
{Brownawell87}
D. W. Brownawell. ``Bounds for the degree in the nullstellensatz."
{\it Annals of Math.}, 126:577--591, 1987.

{CaGaHe88}
L. Caniglia, A. Galligo, and J. Heintz. ``Borne simple exponentielle
pour les degr\'es dans le th\'eor\`eme des z\'eros sur un corps de
charact\'eristique quelconque."
{\it C. R. Acad. Sci. Paris}, 307:255--258, 1988.

{Kollar88}
J. Koll\'ar. ``Sharp effective nullstellensatz."
{\it J. of the AMS}, 1:963--975, 1988.

{KrPa96}
T. Krick and L.~M. Pardo. ``A computational method for diophantine
approximation."
In L. Gonz\'alez-Vega and T. Recio, editors, {\it Algorithms in Algebraic
Geometry and Applications. Proceedings of MEGA'94}, pages~193--254,
Birkh{\"a}user Verlag, 1996.


charles chen escribió en mensaje ...
>I need to know how to prove that n numbers of equations are needed to solve
>equations with n numbers of vatiables. If anyone knows how and is willing
to
>help me out, I would greatly appreaciate the effort.
>
>Sincerely
>
>Charles
>
>
>
>   -**** Posted from RemarQ, http://www.remarq.com/?b ****-
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