I don’t have a mathematical background and am looking for an “intuitively satisfying” explanation (so, for example, the Wikipedia article is way too technical). Perhaps this is not possible in which case, fair enough.
I understand (I think) what a polynomial is and I believe algebraic geometry is about understanding the solutions to polynomial equations using abstract algebraic techniques and geometry. I rapidly get lost when the discussion shifts to rings, fields, schemes and so on. However, I’m not looking to understand all these different concepts but rather get a high level overview.
One day, I’d like to understand how Grothendieck revolutionized the discipline but that may be far too ambitious 🙂
In: Mathematics
Ultimately _algebraic geometry_ is about the application of geometric ideas to algebraic problems. That is a very broad phrasing, but it is the gist of it.
Say you have a bunch of linear equations such as 3x + 4y + 5z = 6. Geometrically the solutions of this one form a plane in 3D, and a general linear equation gives you some arbitrary-dimensional analogue. If you solve all of them, then geometrically you are looking for the _intersection_ of those planes. Even the ancient Greeks already intersected lines and circles, so this is a very old idea.
This allows you to make multiple observations that are less easy to spot when doing just algebra. For example, two parallel planes never intersect and hence there is no solution. Even more importantly we can speak about dimension at all (we already did!). As an actual application: if the planes are not in very specific arrangements (which have probability zero to happen if chosen randomly), then we expect each subsequent equation to reduce the dimension(!) of the solutions by one. So n equations in n variables will typically have 0D solutions, actually exactly one solution (a point, which is 0D!); and n+1 equations in n variables will usually not have any solution at all.
But what I just said does not only apply to _linear_ equations. You can take quadratic, cubic, arbitrary polynomial ones and those statements are still true! A “new” relationship between variables will always decrease the solution’s dimension by exactly one, n such equations in n variables usually have finitely but non-zero many solutions.
This is barely a starting point, though. A good next step would be to look at _elliptic curves_ often written as y² = x³ + ax + b. We can by pure geometry define [“addition” on the solution points](https://upload.wikimedia.org/wikipedia/commons/thumb/c/c1/ECClines.svg/680px-ECClines.svg.png) which actually allows us to classify those solutions among many things; and do cryptography with them.
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