The Geometry of Polynomials (2018) | Hacker News Reader

The Geometry of Polynomials (2018)

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83 pointsvector_spaces24 comments

The whole point of Algebraic Geometry is to describe geometry using algebra. It’s is a very deep, successful and important subject in modern mathematics. I wouldn’t say it needs to recover it’s geometric routes, it’s sort of the whole point. To describe something geometric you actually just need to describe algebras of functions on that object. It’s a very rich subject.

What happens at school and early uni however is a different story. Classical algebraic geometry could definitely be taught to undergrads or high schoolers but there is some weird insistence on only calculus being important. If anything the categorical and algebraic ideas expressed in lots of algebraic geometry follow through into all sorts of other areas of mathematics. I think there are even connections to modern physics. Id say it’s probably one of the more important areas, but it’s relegated to later stages.

Algebraic Geometry is all you say it is, however, working with young engineers who have not had more geometry in their curriculum, I've noticed they don't have that intuitive spatial connection to reality other than through CAD or their lived experience. I am reading Tristran Needham's book, "Visual Differential Geometry and Forms: A Mathematical Drama in Five Acts", and it is amazing. I love Sir Michael Atiyah's famous quote that Neeham uses at the start of the book:

"Algebra is the offer made by the devil to the mathematician. The devil says: I will give you this powerful machine, it will answer any question you like. All you need to do is give me your soul: give up geometry and you will have this marvelous machine."

And it continues: "...the danger to our soul is there, because when you pass over into algebraic calculation, essentially you stop thinking: you stop thinking geometrically, you stop thinking about meaning."

Being someone who worked with their hands in the real world building things before the current Maker movement took off, and then went on to physics, engineering, and more abstract matters, I see this deficit in the young engineers with a Master's in Mech. Eng., but no real-world experience, or intuition of the world around them and geometry. Algebraic Geometry has its place and it has achieved much, but I strongly opine that children should play in the real world a lot, learn mathematical concepts with a strong geometrical underpinning, and then move into things like Algebraic Geometry. I have watched my children's curriculum and it is no where near as heavy in geometry as it once was for me.

I'm sure the reason we emphasize calculus is because it's by far the most applicable part of math to a whole range of subjects. Now days maybe linear algebra is up there as well, but I'd struggle justifying teaching algebraic geometry to general undergrads. I'm sure the applications are there, but calculus just gives you maximal bang for your buck.

Traditonally this was so; today I'd say linear algebra gives more over a larger range of subjects.

Both linear algebra and calculus are classic introductory math classes. A lot of calculus IS linear algebra, given that derivations are linear approximations.

I had three years of high school with a substantial amount of calculus; by contrast while I was introduced to matrices and Gaussian elimination at high school, the treatment of the algebraic side was almost completely superficial and focussed on solving systems of linear equations.

At university that changed, with a clear treatment of vector spaces and linear algebra as generally interesting topics. But still, they received far less time than calculus in my first year in a maths degree.

Now that was over 30 years ago and things might be a bit different today. But I have the impression that the more generally applicable subject of linear algebra gets less time unless students get interested in the applications that demand it be taught properly.

Yea, theorems like least square prove that A^T * A * X = A^T * B. but how exactly is this related to calculus though?

The equations you quoted result from minimizing the square of the norm of the residual of Ax-b over all inputs x, so in a sense least squares is just calculus…

Thanks @peterhalburt33

Transposes correspond to integration by parts, and the question of whether A^T*A has an inverse can get involved. Also for infinite matrices the analogies hold more readily, see the observation due to Alan Edelman on page 8 of https://klein.mit.edu/~gs/papers/Paper5_ver7.pdf

Thanks @selimthegrim.

Derivatives are (derivations are the generalization so of course are but I would be surprised to see an undergrad class treat them)

My mistake. I meant derivatives.

I’d agree, but most first year students aren’t ready to work at the level of abstraction required for a rigorous LA course (i.e., more than just matrices and Gaussian Elimination etc.)

This is 'analytic geometry', not 'algebraic geometry'. There's a big difference!

Yea none of this is “algebraic geometry”

Do people think that algebraic geometry might be thought of as the root of all geometry? (With differential geometry being another root). You've got Euclidean, generalised to affine, generalised to projective, generalised to algebraic geometry. The objects of study in projective geometry are all algebraic sets of some kind; in fact, I'd more accurately say they're subschemes of the scheme RP^n. I think this view has been put forward by Norman Wildberger* (who admittedly has some controversial views on mathematics, but I think this suggestion is helpful for motivation).

* - Without mentioning scheme theory.

> Do people think that algebraic geometry might be thought of as the root of all geometry?

I certainly don't think of it this way.

There are many geometries and spaces that cannot be described fruitfully within the framework of algebraic geometry, much the same as how there are many functions that are not analytic.

You still need algebraic topology to get anywhere useful on continuous functions between spaces.

Maybe a stupid question but I've been brushing up on my calculus in preparation to go back to school for physics.

Is this equivalent to the tangent to the equation at the root? I mean, obviously with exponents higher than 2 you get polynomials not lines, but the given derivation involves just taking the derivative so shouldn't that work? If so, this massively simplifies a bunch of the practice problems I'm doing right now. I should probably just try to prove it

Not a stupid question, and you're definitely on the right track! It's a local linearization in the neighborhood of a root. Because of how simply polynomials can be expressed (they can be defined by a scale and a multiset of roots), it happens that you can read this linearization right off the factored form.

If you're interested, I bet you could look at the behavior of the Taylor series expansion of polynomials in the neighborhood of their roots and see something that provides a good foundation for this article.

n.b. this is somewhat akin to a partial derivative of the polynomal, but rather than wrt to a specific variable, wrt a term that dominates a region. By evaluating the rest of the polynomial with the center of that region (the root), you're just killing the factors that are non-dominant. It wouldn't work if you were a bozo and forgot that $(x-1)(x-1) = (x-1)^2$ and so tried to say it looks like the line $x-1$ locally even though it doesn't locally dominate (the effect of the other $x-1$ is large). Also wouldn't work outside of happy polynomial land because who's to say there's not some $sin(x)$ somewhere that has global effects. If you try to go down this rabbit hole with non-localized effects, you'll encounter the Fourier Transform (and Linear Canonical Transform). If you try to force things to be more localized, you'd probably encounter spectral methods like the Short-Time Fourier Transform as well as wavelet methods. The latter are useful for things like compression because they operate using functions that are "semi-local" but non-trivial.

Fun fact: “proper” polynomials have only non-negative integer exponents but you can construct other non-linear polynomial-like equations with negative and even rational exponents. They work the same with the number of real roots related to the number of coefficient sign changes.

Just don’t expect any general solutions, you have to solve them numerically if they don’t simplify to a common type.

This is gold. I wish they explained polynomials like this to me in school. Italian school system sucks

The Italian school was foundational in this subject, so you might enjoy reading the basic material as they originally wrote it. Just be wary that it's old so the mistakes there have since been fixed.

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