Name: Ideals, varieties, and algorithms
ISBN: 9783319167206

Ideals, varieties, and algorithms

Algebraic geometry is the study of systems of polynomial equations in one or more variables, asking such questions as: Does the system have finitely many solutions, and if so how can one find them? And if there are infinitely many solutions, how can they be described and manipulated? The solutions of a system of polynomial equations form a geometric object called a variety; the corresponding algebraic object is an ideal. There is a close relationship between ideals and varieties which reveals the intimate link between algebra and geometry. Written at a level appropriate to undergraduates, this book covers such topics as the Hilbert Basis Theorem, the Nullstellensatz, invariant theory, projective geometry, and dimension theory. The algorithms to answer questions such as those posed above are an important part of algebraic geometry. This book bases its discussion of algorithms on a generalization of the division algorithm for polynomials in one variable that was only discovered in the 1960s. Although the algorithmic roots of algebraic geometry are old, the computational aspects were neglected earlier in this century. This has changed in recent years, and new algorithms, coupled with the power of fast computers, have led to some interesting applications - for example, in robotics and in geometric theorem proving.

What Hacker News says

WoahNounFeb 2022

Modern Algebraic Geometry is a field that encompasses and somewhat unifies so many different areas of mathematics that it is extremely difficult to distill down quickly. So it really depends on what your currently mathematical background is. If you only know some Abstract Algebra (ring and field theory in particular) and Topology at the undergrad level, it will take a long-time for you to contribute to research mathematics, but you could probably get a feel for the subject after a couple years (shorter if our a graduate student focusing on it full-time). If you don't know anything about those two subjects, it's a very long slog. If you know differential geometry, category theory, complex geometry in addition to the above, you could probably pick it up relatively quickly.

The definitions and machinery make sense if you have enough background, but the why and how we got here is often very unclear.

If you don't care about schemes, stacks and current modern viewpoint of Algebraic Geometry, it's not hard to get a decent understanding of algebraic varieties which are the original motivation in the field. An undergrad book on the subject Ideals, Varieties, and Algorithms by Cox, Little, and O'Shea does a great job of introducing the subject. And has a really cool project on calculating Groebner bases for polynomial equations in sin and cosine to define the configuration space of different types of robotic arms.

thread

spekcularMar 2021

You might enjoy Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra by David Cox.

thread

kaitaiMar 2019

If you do have some math background, "An Invitation to Algebraic Geometry" by Karen Smith et al is a lovely, slim volume that gets to important ideas right away.

"Ideals, Varieties, and Algorithms" is an "invitation to computational geometry" by Cox, Little, and O'Shea. I think parts of it might really appeal to HN readers and it's supposed to be for an undergrad math major audience.

thread

colah3Mar 2015

For the algebraic geometry side, though, *Ideals, Varieties and Algorithms" by Cox, Little and O'Shea is great.

thread

j2kunDec 2014

Here are some great books which are meant to be read (and have not been mentioned elsewhere in this thread, i.e. Axler and Pugh).

Algebra: Chapter 0 by Paolo Aluffi

Measurement by Paul Lockhart

The Nature of Computation by Moore & Mertens

Ideals, Varieties, and Algorithms by Cox, Little, & O'Shea

thread

chaoxuJul 2014

Wikipedia has an article on this, http://en.wikipedia.org/wiki/Linear_equation_over_a_ring it would take a long time to read up the background to understand it, and I don't have the background to distill it to the exact result we want.

Note the wiki article references this book: Ideals, Varieties, and Algorithms by Cox, Little and O'Shea. This is a standard and approachable reference on this kind of topics.

Although this topic is interesting, I think it's best to treating it as a technology. Knowing it's existence and know when to apply them is good enough. Understanding how it works would be a big time sink and sadly doesn't give much useful insights.

thread