Introduction to Algebraic Geometry 2

This article is part of a series of posts introducing the ideas in modern Algebraic Geometry to a general audience, The preceding article is here.

Photo by Mahdi Soheili on Unsplash

So what is Algebraic Geometry about anyway ?

The standard ‘starter’ definition is that it is about ‘the study of solutions of systems of multivariate polynomial equations’

Here is an example of such a system :

and here is another one :

and one last one:

So in that last example, we are asking ‘what can we say about the solutions (x,y,z,u,t) to the system of polynomials listed out?’

Classical AG sometimes restricts itself to studying polynomials with integer or rational coefficients, unlike in that last example where one of the coefficients was Pi.

Actually, in Algebraic Geometry, we specify the sets of solutions we are interested in, adding a constraint such as ‘and where the solutions are rational numbers’ or ‘where the solutions are integers’.

Now, AG asks QUALITATIVE questions about the set of solutions, such as

• ‘are there any such solutions?’
• ‘if so, are there infinitely many ?’
• and so on …

There is a branch of AG that is numerical (called ‘numerical Algebraic Geometry’) which IS interested in finding out the actual numerical values of the solutions, however it is comparatively new (and I know little about it).

Ok so that is the starter definition of AG. Now let me guess what you are thinking at this point …

So AG is all very very boring then ?

Well the thing that makes it not boring (in my opinion), is that AG thinks about these questions VERY deeply, to a quasi-mystical level. Let me try and illustrate what I mean …

Question: who am I, really ?

Consider the following polynomial system, where we are interested in real solutions:

Nothing too complicated going here, right ? Here is a plot of the solution :

a spectacular plot of the solutions to : y=0 in the real plane

Now consider the (real) solutions to the following system:

Nothing too mysterious here either. The solutions are the same as before, y=0, and unsurprisingly a plot of the solutions in the real looks exactly the same, namely :

dramatic illustration of the solutions to : y²=0 in the real plane

So Alain, same solutions for both systems, namely the line y=0. So both systems are the same. Can we just move on now ?

Well no actually. Are you saying that all these systems are ‘the same’ ?

y=0

y²=0

y¹⁷=0

y⁷¹=0

…

They ‘feel’ different though … Something is amiss here.

The Reveal: in fact in AG, the systems

and

are NOT the same, and we will see how we capture the difference (and in doing so, understand in what the difference is, exactly, and why it matters)

Question: what am I, really?

Generally, when we hear the term ‘polynomial’, we think of a function, so for example the polynomial x² is a function f(x) = x², which given a value x returns x².

Yes Alain, a polynomial with one variable is just a function like P(x) = x²+x+1 or Q(x) = x³+3. Different polynomials -> different functions, this is all really very simple, can you just end this post now ?

Well, hear me out just a bit more.

To simplify the illustration, rather than a complicated number system like the real plane, consider a number system with just 2 numbers in it, 0 and 1, where the addition + multiplication * is defined like:

0+0=1+1=0, and 0+1 = 1+0 = 1

0*0 = 1*0 = 0*1 = 0, and 1*1=1

In terms of number systems, it does not get any simpler than this :-)

Anyway, now consider the polynomial ‘x’, in terms of a function P(x) = x, and let us compute what possible values that function can take in our simple number system.

Nothing too complicated here hopefully: If the function is P(x) = x, then P(0)=0 and P(1) =1

Alain, seriously, do you get a PhD for this ?

Well now consider this polynomial : Q(x) = x³+x²+x. Again let us compute the values this takes in our number system :

Q(0) = 0³+0²+0 = 0+0+0 = 0 , and Q(1) = 1³+1²+1 = 1+1+1=0+1=1

So P(0) = Q(0) and P(1) = Q(1)

The functions are ‘the same’, so … are the polynomials the same ? They don’t ‘feel’ the same though. Again something is amiss here …

The Reveal: In AG, polynomials such as x, x³+x²+x etc are NOT seen as functions, and are DIFFERENT regardless of the number system.

Question: what can you tell about me ?

Consider the solutions to the following polynomial system in the real plane: x²+y²-1=0. Hopefully many will know that this is just the unit circle:

remarkable illustration of the solution set of x²+y²=1 in the real plane, ie of the unit circle

Suppose you now just tell me the following :

You: ‘I have this polynomial system with 363682828827 equations in it. I have worked out that for the interval [0.3,0.35] the solution in the real plane is the corresponding portion of the unit circle, ie the arc of the unit circle between [0.3,0.35], like this:

the portion of the unit circle for the interval [0.3,0.35]

Me: ‘The solution set of your system is exactly the unit circle. The end.’ (*)

So without knowing anything about the polynomial equations , and only knowing the solutions on a tiny interval, I could tell ALL the solutions for the ENTIRE real plane.

The Reveal: Much of AG deals with what can be inferred globally from local information.

( * ) well that is the gist of it anyway

Conclusion

the ‘starter’ definition of Algebraic Geometry is that it studies the solutions of systems of polynomials. It asks fundamental , near-philosophical questions about polynomial systems.

In the next post we will start to see that, actually, AG is not just about polynomial systems at all :) Its methods generalise to anything that ‘behaves like’ a polynomial system.

That’s it for now, do let me know your thoughts in the comments!