[HN Gopher] The Geometry of Polynomials (2018)
___________________________________________________________________
The Geometry of Polynomials (2018)
Author : vector_spaces
Score : 82 points
Date : 2022-05-15 08:23 UTC (2 days ago)
(HTM) web link (cre8math.com)
(TXT) w3m dump (cre8math.com)
| fennecs wrote:
| 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.
| karmakurtisaani wrote:
| 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.
| chalst wrote:
| Traditonally this was so; today I'd say linear algebra gives
| more over a larger range of subjects.
| peterhalburt33 wrote:
| 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.)
| auggierose wrote:
| Both linear algebra and calculus are classic introductory
| math classes. A lot of calculus IS linear algebra, given
| that derivations are linear approximations.
| selimthegrim wrote:
| Derivatives are (derivations are the generalization so of
| course are but I would be surprised to see an undergrad
| class treat them)
| auggierose wrote:
| My mistake. I meant derivatives.
| sshlocalhost98 wrote:
| Yea, theorems like least square prove that A^T * A * X =
| A^T * B. but how exactly is this related to calculus
| though?
| selimthegrim wrote:
| 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
| peterhalburt33 wrote:
| 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...
| chalst wrote:
| 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.
| eggy wrote:
| 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.
| bigbillheck wrote:
| This is 'analytic geometry', not 'algebraic geometry'. There's a
| big difference!
| ABeeSea wrote:
| Yea none of this is "algebraic geometry"
| kloch wrote:
| Fun fact: "proper" polynomials have only non-negative integer
| exponents but you can construct other non-linear polynomial-like
| equations with negative, rational, and even real 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.
| ogogmad wrote:
| 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.
| OccamsRazr wrote:
| > 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.
| ABeeSea wrote:
| You still need algebraic topology to get anywhere useful on
| continuous functions between spaces.
| fedeb95 wrote:
| This is gold. I wish they explained polynomials like this to me
| in school. Italian school system sucks
| Cobord wrote:
| 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.
| wholinator2 wrote:
| 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
| alhirzel wrote:
| 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.
___________________________________________________________________
(page generated 2022-05-17 23:02 UTC)