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Systems of polynomial equations are central to mathematics and its application to science and engineering. Their solution sets, called algebraic sets, are studied in algebraic geometry, a mathematical discipline of its own. Algebraic geometry has a rich history, being shaped by different schools. We quote from Hartshorne’s introductory textbook (1977):
Algebraic geometry has developed in waves, each with its own language and point of view. The late nineteenth century saw the function-theoretic approach of Brill and Noether, and the purely algebraic approach of Kronecker, Dedekind, and Weber. The Italian school followed with Castelnuovo, Enriques, and Severi, culminating in the classification of algebraic surfaces. Then came the twentieth-century American school of Chow, Weil, and Zariski, which gave firm algebraic foundations to the Italian intuition. Most recently, Serre and Grothendieck initiated the French school, which has rewritten the foundations of algebraic geometry in terms of schemes and cohomology, and which has an impressive record of solving old problems with new techniques. Each of these schools has introduced new concepts and methods.
As a result of this historical process, modern algebraic geometry provides a multitude of theoretical and highly abstract techniques for the qualitative and quantitative study of algebraic sets, without actually studying their defining equations at the first place.
On the other hand, due to the development of powerful computers and effective computer algebra algorithms at the end of the twentieth century, it is nowadays possible to study explicit examples via their equations in many cases of interest. In this way, algebraic geometry becomes accessible to experiments. The experimental method, which has proven to be highly successful in number theory, now also adds to the toolbox of the algebraic geometer.
As in other areas of pure mathematics, computer algebra may help
to discover unexpected mathematical evidence, leading to new conjectures or theorems, later proven by traditional means,
to construct interesting objects and determine their structure (in particular, to find counterexamples to conjectures),
to verify negative results such as the nonexistence of certain objects with prescribed invariants,
to verify theorems whose proof is reduced to straightforward but tedious calculations,
to solve enumerative problems, and
to create data bases.
There is a growing number of research papers in algebraic geometry originating from explicit computations. The computational methods also play a significant role when it comes to applications of algebraic geometry to practical problems. And, they enter the classroom, allowing us to introduce students at an early stage to algebraic geometry, without developing too much of its abstract machinery.
Introductory Remarks on Computer Algebra
Basic Notations and Ideas: A Historical Account
Basic Computational Problems and Their Solution
An Introduction to SINGULAR
Homological Algebra I
Homological Algebra II
Solving Systems of Polynomial Equations
Primary Decomposition and Normalization.
Algorithms for Invariant Theory
Computing in Local Rings
A Sheaf Cohomology and Beilinson Monads
B Solutions to Exercises