Intuitive descriptions for the principles of divisor and also category

When attempting to clarify AG-codes to computer system researchers, the significant factors of opinion I am confronted with are the principles of divisors, Riemann-Roch room and also the category of a function area. Exist any kind of instinctive descriptions for these principles, ideally descriptions that are much less depending on expertise of algebraic-geometry/topology?

2019-05-07 10:41:30
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Answers: 4

If your CS close friends resemble me, they could still locate the solutions over a little frustrating. So you can start as adheres to:

First, show them the Wiki's divisor web page (it constantly functions!). After that clarify to them that by the fundamental theorem of arithmetic, any kind of divisor is simply a number of prime numbers with multiplicities.

Next, inform them that entirely comparable to $\mathbb Z$, the FTA persuades $\mathbb C[x]$ (which you can take a line). Other than since each "divisor" (polynomial) can be taken a number of factors (origins of the polynomial) on that particular line, with multiplicities.

Yet why stop as a straight line? One can do the very same point for a (sensibly wonderful) contour on the aircraft, and also an all-natural means to get a number of factors is to converge with an additional contour. Incidentally, a number of factors "separate" the contour, more warranting the terms!

Back to $\mathbb C[x]$, you can mention that the complete varieties of factors (with mult.) is the level of your polynomial $p(x)$, or the measurement of the vector room $\mathbb C[x]/(p(x))$.

Now, if they still follow you, show them Matt E's fantastic solution (-:

2019-05-09 10:53:15

Well, the principles of category and also divisors originate from geometry/topology intuition initially, so those descriptions are usually mosting likely to be the fastest.

Nonetheless, we can offer really harsh variations without requiring to claim "category is the variety of openings in the surface area that the contour resembles." You can consider category as an action of just how difficult the function area is. Normally $k(x)$ is mosting likely to be the most basic feasible one, and also it takes place to be genus absolutely no. Category one function areas resemble $k(x,y)$ yet where $y^2$ is a cubic in $x$. Currently, this isn't really specific, yet it's a harsh point.

When it comes to divisors, the topology/geometry of them is "straight mixes of factors, modulo absolutely no - $\infty$ loci of features," yet to manage it totally algebraically, you'll intend to transform words "factor" to "distinct evaluations", so these are integer straight mixes of the numerous means to define the order of a component of the function area, modulo some set of unimportant ones.

2019-05-09 06:58:57

Charles has actually currently clarified the idea of category (which originates from geography - - there's additionally a totally algebraic idea of category as a first cohomology team, yet I at the very least would certainly locate it much less instinctive). So I'll speak about divisors.

First off, the analytic idea of a nonsingular projective contour is a portable Riemann surface area. "Nonsingular" translates to "intricate manifold," "contour" converts to "measurement 1," and also "projective" translates to "portable." So I will certainly speak about Riemann surface areas. *

If you have a Riemann surface area, it's in your area the like the facility aircraft. And also in the facility aircraft, you have a means to gauge the order of the absolutely no of a holomorphic function. Extra usually, you can gauge (in $\mathbb{Z}$) the order of an absolutely no of a meromorphic function (which is adverse if it has a post, absolutely no if it is nonvanishing and also analytic, favorable if it has an absolutely no). So to any kind of meromorphic function (certainly, we do not limit to holomorphic ones - - - they're all constant) on a portable Riemann surface area, we can assign a limited set of factors with multiplicities containing the areas where the order is nonzero (i.e., the absolutely nos and also posts).

A divisor is extra basic. It's simply an official amount of factors on the Riemann surface areas with multiplicities. A divisor might originate from a function as above ; after that it's called principal. As a whole, nonetheless, it does not. Yet offered a divisor $D$, we can associate a vector room of meromorphic features such that $div(f)+ D$ is a divisor with nonnegative coefficients. This is represented $L(D)$ and also its measurement by $l(D)$.

The Riemann-Roch theorem is after that a declaration concerning the measurements $l(D)$. Among its effects, specifically, is that if $|D|$ (the amount of the multiplicities) is huge, after that you can constantly locate a (nonzero) component of $L(D)$. This is with ease understandable. If $D$ is huge, after that you are permitting great deals of flexibility for $f$, possibly having posts at several areas.

*I'm being casual below, yet there is a basic "GAGA" concept concerning these equivalences that enters into far more information, which I recognize nothing around.

2019-05-09 06:58:46

If you intend to stay clear of algebraic geometry and also geography, after that you will possibly be compelled to make use of algebra. This is all right, due to the fact that things you inquire about have algebraic analyses.

First off, it possibly aids to write the function area as $k(x,y),$ where $x$ and also $y$ are connected by some formula $f(x,y) = 0$. One trouble is mosting likely to be that $f$ can not be picked to be smooth as a whole (due to the fact that not every contour can be installed as a smooth aircraft contour), yet it's possibly best to overlook this ; if it comes to be vital, after that possibly your coworkers will certainly be diving much deeper right into the concept anyhow, therefore the entire degree of description can be increase.

Currently to clarify divisors, you can have them visualize converging $f(x,y) = 0$ with a few other contour $g(x,y) = 0$ ; the junctions will certainly be a number of factors, perhaps with multiplicity. This is a divisor. So divisors are simply all-natural means of inscribing just how contours converge each other. (Again, I am overlooking below the concern of factors at infinity ; you will certainly need to determine whether that is ideal, or whether you require a greater degree of accuracy in your descriptions.)

When it comes to the category, it's a little bit extra difficult to clarify algebraically, yet feasible ; below goes:

Suppose that the formula $f(x,y) = 0$ has level $d$, so your function area represents a level $d$ contour in the aircraft. Currently allow $V_n$ be the vector room of all polynomials of level $\leq n$ in $x$ and also $y$. Allow's intend that $n \geq d$ (and also as a whole, we need to assume that $n$ is huge).

A straightforward calculation reveals that $V_n$ has measurement $(n+2)(n+1)/2$. Inside $V_n$, we have a subspace containing all the multiples of $f$. This subspace is gotten by taking the components of $V_{n-d}$ (i.e. polynomials of level at the majority of $n-d$) and also increasing them by $f$, i.e. it is the subspace $f V_{n-d}$ of $V_n$, therefore has measurement $(n-d+2)(n-d+1)/2$. If we consider the quotient $V_n/fV_{n-d},$ this after that has measurement $n d + 1 - (d-1)(d-2)/2$.

What is the definition of this ratio?

If $g \in V_n$ stands for a non - absolutely no component of $V_n/fV_{n-d}$, after that it is a level $n$ formula that is not divisible by $f$, so it does not disappear identically on $f(x,y) = 0$, so by Bezout's theory, the junction of $g(x,y) = 0$ and also $f(x,y) = 0$ is $n d$ factors, i.e. a divisor of level $n d$. So we see that non - absolutely no components of $V_n/f V_{n-d}$ represent those divisors of level $n d$ that are gotten by converging $f(x,y) = 0$ with a level $n$ contour. (In reality, we need to think of non - absolutely no components approximately scaling, due to the fact that if we increase $g$ by a non - absolutely no scalar, the contour $g(x,y) = 0$ does not transform.)

Hence amongst all the level $n d$ divisors on $f(x,y) = 0$, those that stop by converging with a level $n$ contour create a room of measurement $n d - (d-1)(d-2)/2.$ (Here I have deducted 1, due to the fact that rescaling make up 1 of the measurements in the above formula.)

Currently what is the measurement of the room of all divisors of level $n d$ (with non - adverse coefficients, which are the only kind that can perhaps emerge as junctions ; divisors with non - adverse coefficients are called reliable)? Well, we simply need to pick $n d$ factors and also add them with each other. We are picking the factors from a contour, which is one - dimensional, to make sure that suggests there is an $n d$ - dimensional room of reliable divisors of level $n d$.

Hence we see that, in the $n d$ - dimensional room of all reliable divisors of level $n d$, those that emerge by converging with a level $n$ contour are a subspace of measurement $n d -(d-1)(d-2)/2$.

The amount $(d-1)(d-2)/2$ is specifically the category. So what we see is that the larger the category is, the tougher it is for a level $n d$ divisor on $f(x,y) = 0$ to be gotten by converging with an additional contour.

As an example, if the level $d$ is $1$ or $2$, after that the category disappears, and also every level $n d$ divisor originates from converging with a level $n$ contour.
E.g. on a conic (i.e. when $d = 2$) any kind of 2 factors originated from converging with a line (the line that travels through those 2 factors), any kind of 4 factors originated from converging with a conic, and more.

On the various other hand, if the level $d = 3$, after that not all triples of factors on $f(x,y) = 0$ originated from converging with a line : actually if you offer on your own 2 factors, after that they establish a line, which subsequently establishes the 3rd factor of junction. In a similar way (rising to the instance $n =3$), a basic set of 9 factors on the cubic does not originate from converging with an additional cubic ; rather, if you offer on your own 8 factors on the cubic contour, you can locate an additional cubic travelling through these 8 factors, and also its 9th factor of junction is distinctly established by the offered 8 factors. (Since 9 = 8+1, this is a certain indication of the reality that our cubic contour has category 1.)

As you can see, I've expired right into an extra geometric mind-set (making use of ideas such as measurement), yet I do not assume one can prevent this entirely : the principle of category emerged traditionally from basically the sort of calculations that I've simply been making, and also at some time you need to think of rooms of divisors and also their measurements if you intend to recognize it. Still, I wish that this offers you a method to clarifying the principle which is extra algebraic, therefore extra obtainable to your coworkers.

Another technological statement : The room $V_{n}/f V_{n-d}$ is an instance of a Riemann - - Roch room, and also the formula for its measurement ($n d + 1 - $ the category) is a grandfather clause of the Riemann - Roch formula.

[Technical statement included feedback to an inquiry of T. in the remarks listed below ; do not hesitate to overlook it if it goes to expensive a degree : ] Note that the above conversation functions also if $f$ is permitted to be single (and also irreducible, claim, to make sure that we can not have polynomials $g$ which disappear on one part of $f(x,y) = 0$ without disappearing overall contour). The factor is that $(d-1)(d-2)/2$ is constantly the math category of an aircraft contour of level $d$, whether the contour is smooth, and also it is the math category that interferes in the Riemann - - Roch formula. (I think that this is the beginning of the adjective math in math genus : it is this variation of the category for single contours that shows up when you make Riemann - - Roch - type estimations of the measurements of numerous rooms of divisors.)

2019-05-09 04:26:14