# What are the applications of algebraic geometry to electronics?

I am currently pursuing an undergraduate degree in Electronics and Communications Engineering. I recently got a research internship to study algebraic geometry for two months. I would like to know what exactly the applications of algebraic geometry are, in the field of electronics and communications, signal processing, control theory and other areas. In short, I'm curious to know about the real-world applications of algebraic geometry.

It is difficult to say anything not too generic but the very typical problems that appear in engineering are:

1) Can I decompose this complicated problem into easier to understand problems? 2) I understand simple relationships. Can I extrapolate these simple relationships into solutions of more complicated systems?

You've likely seen 1) as an electrical engineer. You always try to take complicated functions, such as a triangle wave, which are hard to create in nature and use your basic functions like sine and cosine to reconstruct them.

Maxwell's equations are an example of 2) but even more fundamental is Newtonian mechanics. It is easier to understand that velocity is linear than to say that a falling object has position given by a parabola. It is easier to understand that the time-change in a current in an inductor induces a proportional change in voltage than to see that the voltage is an exponential.

Back to your original question, what does this have to do with algebraic geometry? It turns out that many of the questions asked in these contexts a) extend to the complex numbers b) extend to projective spaces.

With appropriate adjectives inserted, analytic questions on projective spaces over the complex numbers can be phrased as algebraic questions via Serre's GAGA theorem. Writing the triangle wave as a sum of sine and cosine functions, while on the surface looks strictly analytic, actually can be deduced by working with algebraic functions on the projective complex line.

In computer-aided geometric design (CAGD), algebriac geometry is used to solve implicitization and inversion problems.

For some background, see here or here.

The results used in CAGD are rather ancient (I guess "classical" is the polite word), but they are from the field of algebraic geometry, nonetheless. I don't think modern algebraic geometry has applications in CAGD. But I can't understand any of the modern results, so I might be wrong.

CAGD, in turn, is the basis for many "real-world" geometric computations in engineering and manufacturing, though these are a bit removed from the applications you listed.

See also this question, though the answers given there are again not in the areas you mentioned.

• Groebner bases are not so ancient are they? Feb 25, 2014 at 14:36
• @Steve: No, Groebner bases are not so ancient. But the applications I had in mind are mostly based on resultants and elimination theory, which are more than 100 years old. Feb 26, 2014 at 1:31