PSE Lectures Ep 23 - Rational functions on elliptic curves. Zeros & Poles

Transcript

hey guys so Carlos told me

that you will be late in five minutes um and we can start without him but uh are we waiting for uh more people ER maybe meanwhile er do you have any

questions about the previous material okay so so let's start and uh I think Carlo should be here in in a few minutes okay

so um I want to take a step back from ER rational functions and er it talk a bit about the basic properties of the the ailan group of rational points on elliptic curve so uh the first thing maybe I mention it but it's good to to recall um for an elliptic curve over field

K ER if if I have a field extension um I mean of of K then the K rational points are a a subgroup of the K of the L rational points of E I mean

when you look at it E when I say e over K it means that the the elliptic curve is defined over K um so you you can write y sare equal x CU + a x + B with um A and B and K so of course if it isn't defined over over k then for any field extension l i can consider I mean formally it's it's it's a different object it's it's it's the same the same equation but

considered over the bigger field L and the K rational points are simply I mean the solution for this equation so ER clearly you have we have a a a subset inclusion of the K rational points into the El rational points but I claim that this subset inclusion is is in fact a sub group inclusion and the reason is that the the operation that we have in in L for example it restricts I mean if you take

a a k rational point in A in E of K or maybe if you take two then doing p+ Q is is solving a bunch of equations that are defined over over K and also the solutions of these equation are um are elements in K so when you consider a a p and Q here and you do p plus Q you're doing the same

so this is why we have an actually a subgroup inclusion is this um is this point clear this this is important okay so that's the first thing the second thing ER what we will consider ER

mostly ER is is the case where K is a finite field so let's say it's fq for some uh some Prime power uh q and then we have an elliptic curve defined over fq and remember that we have a a kind of a partial order set of all field inclusions I mean here at the

bottom you have FP this is included in in in all the bigger ones so in particular in fq and fq is included in for example fq s but it's also included in F QQ but here we don't have a field inclusion H however if you go high enough in this case uh

F Q to the 6 then you you do get a field and and and now I I mean of course at the end of this chain sits fq bar which I I remind you I mean there are many ways to H to write it I write it slightly informally as the union of all fq to the N this is a this is a bit of a lie ER

because ER I mean as I said it is f fq to the I mean the union of all these things but not all not all of these pieces are sub subgroup inclusions ER with the next one so you need to be a bit more careful but in any case um I have this uh this partial order set of field inclusions and I can simply apply e on all of that and I get that the FP rational points are

are a subgroup of the fq rational points which are for example a subgroup of the F q s rational points and so on and this is all contained in the F Q Bar rational points now so this is a tower of a a sub billion well a

inclusions now of course each EF Q to the n in the fq to the N rational points over e ah I I'm sorry one thing here is a is um is something I cannot do um and this is this f e f P can anybody say why why I cannot consider

efp I mean the FP rational points over e remember I have a I have a field K the field k for for us in this is fq or let's say n is bigger than one so fq is strictly bigger than FP and I consider e over over K so e fq if I consider e over fq I cannot I

cannot talk about um the the FP rational points over e I'm sorry because of Q because of Q will be minimum because um I mean F Q let's say n is bigger than one so so Q is a p to some actual power it's not it's not just P then when I talk about when I say that I have an elliptic curve defined over fq it means that the the coefficients of the equation A and B are in

fq and potentially they could be in fq but not in FP right because fq is strictly bigger than FP and then I cannot talk about solutions to the equation over FP I mean it simply doesn't add up okay so that's that's an important point I mean if you you have an elliptic curve defined over fq or defined over any field you can only consider rational points on a field extension but not on a on a smaller field H okay so this this was a this

should be removed but yeah yeah sorry I I think I I feel I need to ask this now before we go any further uh I looked again at your notes I didn't look at the video from last week I L your notes and there's not a clear definition of what you call CLE rational points from one maybe I missed it but you always mention care rational points for me you're just with I just make the the shortcut of just picking points on the but it's more complex than this and so I quickly remind the it's so H let me put it here e k this this is what I call K rational points this is

the set of solutions X comma y right such that y² = X CU + a x + b a X and Y need to be in case in in K square right but why why is the link with rational function like why are we calling them rational points no we the the link to the let me just complete this and we also have the point at Infinity yep okay H the reason for the name is that

um the the the main case people typically consider a a solution to elliptic I mean elliptic curve points is over the rational numbers okay ER so by you know by extension people call it K rational points or just K points yeah okay okay Max but we are talking about the points on the eliptic curve we Infinity right so it's like the K rational points are points on the elliptic curve with coordinates in the field perfect thank you okay so okay so I hope the rest of what I

said is is is clear and then each of the the the group of of fq to the N rational points is finite right and this is by the way a reason that um we did the classification of finite a billion groups I mean now that we have the classification we know that any any of these groups has a a canonical form right as a product of cyclic groups or

direct sum of cylic groups but um I claim that e fq bar is never find it so why is that ER so take assum a y z in fq bar then I I can consider the the the

polinomial h x Cub + a x + B minus y0 sorry minus y0 squ y z is now just a constant and this is a polinomial let's call it f h this is a polinomial over fq bar right and over f Q Bar every polinomial

has a root so there is some x0 in fq bar as such that F of x0 is zero and this means that the point x0 y0 will belong to the K rational points sorry to the F Q Bar rational points on E and you see I didn't I took any

arbitrary y z with no restriction and I found a point so since since fq bar is infinite and so does so is e of fq bar okay so that's the third remark the fourth remark is um

we want to understand I mean each of the the finite groups this say the yes can I ask about the proof um isn't it a bit like uh um like uh there is infinite uh number of uh x0 and one y z that solves uh the equation yeah because for every y zero there is an x0 the and there are infinitely y

infinitely many y zeros because it's fq bar yes but there is infinitely many y zeros because the fq bar is infinite yes and because there is infinite uh number uh it I don't know it seems like it is like uh they they refer to each other number infinite number of Y zeros and the Infinity of fq bar uh refers to each other so I can't understand how we make conclusion so what I need to show you is that there are infinitely many solutions to the

equation yeah but if I show you this then I prove my claim right yes so now I I I fix y z for every y Zer that I fix I find the solution I find the I find a solution for a for I mean I find the root for fee right yes but can it be the case that a a Q Bar is infinite but there is not infinite amount of solutions or it depends on the equation I'm I'm trying to no no it doesn't depend on the

equation it's always true so I'm trying to to explain I for every y Zer I can find a root of P right right a root of Fe is some x0 uh in fq bar right that nullifies P but if you then take x0 comma y z then they the they they constitute a k rational point I mean an fq bar rational Point all right all right this makes sense okay and I can do it for for any y

z i I can find x0 such that x0 and y z is is a is a fq bar rational point they are all distinct points because if you take y0 and y0o Prime that are different then X even if the x0 is the same then the y z and y z Prime are are different so so right so if you take a y z Prime H potentially well it has some I mean then Fe has some root x0 Prime potentially x0 Prime is is the same as x0 but I don't care as long as y z is

different than y z Prime then these two points are different yes right right makes sense thank you okay so so if you give me infinitely many y zeros and I I am over fq so I can always find a root to to Fe then I can find infinitely many points on the on efq okay Lisa yes yes yes thank you okay all right so

um what are in the points of of order two in a e of fq remember for an Aban group a point of order two is a point that if you add it to itself you get a you get zero Nicholas you I can try to read but no no sorry I was just reacting to it's

it's okay no it doesn't have any meaningful yeah ER er a point of order two it means a a some p in fq bar sorry in E of fq bar such that 2 p is equal to zero now remember the we had a definition when H to how to how to define H P plus p

um to Define p plus P we need to take tangent this this is in the the the real case we need to take the the tangent to p and it in intersect with another another point on the curve just one other point we call it P star p and we reflect we get the definition of p plus P

now to say that 2 p is equal to to O it means that this point P star P or rather this point is the point at Infinity so 2p is equal to zero um if and only if ER the tangent to e at p is very vertical and these are precisely the

points here and potentially if I had like a another Circle then these two points are are also with a vertical tangent line right and then it interest I mean if you take a P here then the tangent line intersects only the point is infinity you reflect you get a you get still the point at Infinity you get that P plus p is equal to yes one question so the tangent needs to be vertical at P or at 2 p it's p and then and then 2p will be

zero ah okay okay now thanks thank you okay we are looking I I want to I want to understand this is the the the kind of baby case I want to understand all the points of order two so I claim that points of order two have a a vertical H line they must have right because when you do p plus P you take the tangent line and you look for the intersection of the tent line with with the Curve if if it intersects in in an actual point on the curve not the point at Infinity in in like a finite

point then two I mean p plus P will never be zero the only way that P plus P will be zero is if the tangent line doesn't intersect the curve I mean doesn't intersect the curve at a finite point and then we say that it intersects the curve at at the point at infinity and and uh and in and then 2 p is defined to be uh to be o this is of course this is only for the the the the drawing is only for psychological reasons I mean whatever I said here is Meaningful over any field and we we indeed Define the tangent line and what is a vertical

tangent line over any field um so 2 p is equal to zero if and only if the the y coordinate of p is zero okay that's that's the only way that the tangent line at P will be H will be zero now I put it in the equation and I substitute a y equal Z so

um x0 is a root of a what we called s or S of e this this cubic polinomial X Cube plus a X+ B over of course over the field we work I mean we now actually we look for points of order two over fq uh

so you you look for I mean the points of order two they they must satisfy I mean they must be of the form x0 comma 0 where x0 is the root of a of this polinomial over fq bar and we know that over over fq bar this polinomial s has exactly three Roots maybe we call it whatever um write s e x equals um x - x0 x -

X1 x - X2 and then and the points of order two this is denoted as e bra brackets 2 this is the set I mean of course the point at Infinity is of order and um we have three others which are x0

comma 0 X1 comma 0 and X2 okay now in fact E2 is a subgroup of efq bar because if you you add two points of order two you get another point of order so we can ask I mean this this is a

group in itself and um by by the groups we we know that it can be isomorphic um just two there are just two options one of them is of course integers modul 4 there there are simply only two two a billion groups of order four one is is H

integers modu 4 and the other is integers modul 2 direct with integers mod okay and in um we will see that um that E2 is a is the latter is isomorphic always

to to this group this is called by the way the decline yes sorry one question can you repeat what exact is a point of order to I think I didn't get that so this in general for any group in an element in a group the order of the element is the minimal integer the minimal natural number ER such that if you ER if you add the point to itself this many times H you get

zero ah okay okay so points of order two is simply is the the points that if you add them ER I mean to themselves two times you get zero is that the characteristic of a fe no I mean the character okay you can say that if you are given a field you can look at it a you can extract from it the a billion group with respect to addition and then the characteristic of the field can be defined as the order of one ah okay okay so because the

characteristic of the field is the minimal number such that if the minimal integer n such if you do one + one + one n times you get zero I see okay thanks so yeah but more generally we we will want to consider the the points of order n in F fq Bar sorry in E of fq points of order

n now note that of course I can consider I could consider points of order two in one of the the intermediate finite groups right I mean I can always consider the collection of points of order two in any in any ailan group but in fq bar I get the complete picture right I mean here in this this um this drawing that is over R it could be that this circle does not exist and then I don't have a I I mean I don't have these

points on the Curve but if I go to the the algebraic closure of R namely the the complex number I will have a I will always have two other points so it is it is nicer to ER to look at points of order two in the kind of the ideal setup that's and that's the the algebraic closure uh okay so this this is just a little bit about the the structure the the group structure and the things related to

it um are are there any questions about the the basic group structure any of of points all right so last time we talked about um two things we said um so again the setup is e an elliptic curve defined over field K and we talked about pols

we denoted it as K square brackets e this is a pol this is what we want to think about as pols over e h that is I take pols in two variables and I mod out by a by the polinomial Fe let's say Fe is X Cub + a x + B minus y^ 2 and to say that I mod out I told you

that it is it is analogous to er um mod I mean mod Fe arithmetic that we had in in one variable but the analogy you should take it with with a grain of salt because um in in the case of one variable in let's say KX over a polinomial fix H we could do we could do like um actual long division I mean to to to find out

what is um what what does a polinomial FX looks looks like in in the qu in the quotient KX over FX we could do a long division I mean we could say um okay FX is some Q Fe plus r and and then FX is in fact equal to RX in

in the qu k x mod ER and and when we talked about equivalence relations we said that this is the same as saying that um we can define an equivalence relation on a pols with one variable by saying um f is equivalent to G if and only

if F minus G well if if divides F minus G and of course to say that F divides F minus G or actually if I want to keep the notation I should say F minus r right and and you can see that if I have long division then then to do long division is the same as to do I mean to Define this equivalence relation and to mod out by the equivalence

relation now in the case of polom in two variables I don't have long division I mean there are some some candidates for for long division but you never get uniqueness uniqueness of the remainder and uniqueness of the of the quent so er it is not it's it's not well defined and I I I here in the case of two variables I could not do this one but equivalence relations are a much more General notion and they do apply here so when I say

k is a KX y modul Fe I really mean that I take kxy times itself and I mod out by an equivalence relation which is fxy is equivalent to rxy if and only if Fus R is divisible by Fe and divisibility is is a notion that does apply

for for polinomial into variables right you you can say for example that xus y divides x - y * x + y okay so this is just a just a warning H you cannot we cannot do long division in pols in two variables but we can still Define an equivalence relation and this is the the analog of the one variable case Okay so this is polinomial in e now of course

um we can view so if um let's say if K is equal to R but no um if f is a polinomial over e well if it's a polinomial over over k then it defin the function from K squ into K right you you simply you take you take a point in K Square you you substitute it in the polinomial you get a point in you get a an element in

so if if K is equal to R you can really view a polom in two variables as an actual function that goes from the the two-dimensional plane to the the onedimensional line now if I have a [Music] um a polom in K of E I can still view it as a function in fact I can view it as a

function um just a minute I can view it as a function that is defined over defined on E so if you have this this curve then now I can I can simply look at the polinomial is is giving giving value only on the on the purple line and then you get some a bunch of points here okay so

so in general of course this does not depend on the fact that we are in R F in K of e defines an actual function from um the finite K rational 2K simply by by substitution yes there was a question

Nicholas yes sorry um uh a little bit above you think we can always Define a function on two variables to be then then the to go on to K and not K Square I mean k of X and Y but is it always true like can we always substitute one variable with like for y sare + YX uh you still it's like a general result we can always substitute one for your I can always express y equal something in only X and substitute in in the original pols I'm I'm surprised by this just I mean I

say that I'm not sure what you you're asking but if I have a a a polom over e this is I mean I can interpret it my claim is that I can interpret it as an actual function from E this is e or the the the finite er er rational points on E into the the ground field how do I do this okay I have this this h i mean I take a point p and I plug it in I take F of right this I can always do I mean p has

some coordinates x x0 y z you just evaluate it yeah I just evaluate okay so I can and the equivalence relation on K of e guarantees that if f is equivalent to G then and and P is is a rational point over e then F of p is in fact equal to to G of P okay so this this definition does not depend on the representative that I choose okay okay ER and okay so so this is this is for the

the polinomial right you so in other words a general polinomial in two variable it gives you a function from the whole aine the whole aine space in this in this case and this would be R2 into the the ground field and a a a polinomial over e is you can think of it as like a restriction of a polom on the Aline plane restricted to the the curve the Aline well the Aline part of the curve I mean all the finite K rational points of the

curve okay so these These are actual function I mean you can think of them as actual functions although much much of what we do is treating them as as a a formal I mean formal symbols now we also Define um the rational functions on E the rational functions is I I take now K of

e product with itself but in the second coordinate I do not allow the zero polinomial and I mod out by an relation that is defined is I take a pair FG and I say that it's equivalent to F Prime this FG should be should be viewed as what I want to consider as as f modu g and and this is equivalent to to FR Prime G Prime if and only if this this the

f * G Prime is equal to to f Prime G but this equality again this is in K in pols over e right so in other words if you want just to look look at them as polinomial in two variables then there are equal in K of e if and only if FG Prime minus F Prime G is divisible by okay so this is this is the the rational

functions on E and the rational functions on E we said that it's not exactly a given given some Ral rational um we we don't have an actual an actual function from ER from the aine plane to ER to the ground field the ground field is just the aine

line so this is sometimes denoted as dotted doted Arrow okay so this is this is like a a partial function but um what are my problems my problems are um I mean r is a regular a yeah actually here I wanted to say of course we do rational functions on e

that's like partial functions on the K rational points and a given rational function is regular at P if a um there exist a representation r equal f mod G such

that a g of p is non zero and of course for the the I mean if I take a rational function R and I know it is regular at some point I mean I I know is regular at a point P then I can Define um R of p is simply you take this this representation for which the denom denominator does not vanish and you define RFP as as this as

this quotient it could be zero but I mean if FP vanishes then it would be zero but it would always be defined always be a number I mean an element in k um but now we can say if R is not regular then um I simply Define r of P to be Infinity and this gives me an actual function which I also call R it goes

from the finite rational points now instead of the Aline line I now take the projective line the projective line I remind you is just K I mean one way to represent it is just k plus an additional point at Infinity okay so so this is a way to H to extend the definition to get an actual

function but H what we're missing is how to define the function at the the point infinity and this is what I started to talk about last time but I I will get back to it this time so before we take a break uh any questions on this yes um maybe I misunderstood the word fngr because I'm surprised R of p is in K I mean r of p is just uh is two coordinates right R is a is a polinomial

in two variables yeah and P is a is a point like a finite point on E so it has two coordinates X x0 and Y Z well yes so just evaluate it and you get one call one element in K yeah and my other question is on the project staff I admit I didn't follow this part how do you define the projective yes so the projective line um P1 it should should be denoted P1 K it depends on the the field you take the set of I mean you take a k square and you mod out by an equivalence

relation that says that X Y is equivalent to Lambda X Y or Lambda X Lambda y for any non zero Lambda in K and the the equivalence class you we denote as X doy X col Y where of course it I mean sorry here I should have said it's like

the it's K squ minus the origin modul this equivalence relation and we represent a point at on the projective line as having two coordinates X Y but you can I mean since I removed the the origin it cannot be that both X and Y are zero so if a um if h y is non zero We call we can say h i mean if Y is than

zero then X Y is going to be equal to X over y 1 right yeah and by that you get all the points in K okay and if ER Y is equal to zero then um then I get one point I mean X comma 0 is going to be equal to H to X Prime comma 0 for any any non zero X

Prime no okay so what I'm saying is these are called the the finite points or the aine the aine or finite points in the the projective line and then I have just one additional point which I call this is the point at Infinity so I can represent the projective line as as just all the elements in K plus an

additional Point Infinity wait why X the last X zero is the point I thought I thought you could say you can list out all the X's because no but you will get equality here because I mean if x and x Prime are nonzero elements then there exists a Lambda yeah that I mean they defer by by a factor of Lambda for some Lambda sure but why is the point Infinity now so this is just a terminology but it is I'm saying like here I get a collection of points I mean when Y is is different

than zero yeah I get I get a collection of points that can be that is in one to one correspondence to all the elements in the field y okay and in the case y equals z i get only one point there there could be only one point with a a with the the y coordinate equal to zero okay so you're you're defining it as the point infin yeah so I call it the point at Infinity yes okay and this is compatible with this homog homogenization of pols and that we discussed I mean essentially what we're doing here

A1 can be embedded in in P1 by sending a point x into X col one like this is there the Aline representation of the of the point in P1 and this this doesn't cover of course this this doesn't cover the the point at Infinity okay so so in fact I can I I can look at a rational function as as an actual function but when I I modify the the range to be instead of k h to be P1 but but to get the complete picture I

would want to define the rational function on the point O Okay because this is the the kind of natural thing to do if I any way modify the the the range to be projective then I I might as well modify the domain to be the projective I mean the the projective version of the curve namely the the one including uh the point at okay so let's take a 10 minutes break and we H come back to Define

this um can we start

again yep let's go all right so we're now in the business of um taking a a a rational function on e um we want to define the value of r at the point O the the point at Infinity so I mean I can of course I

can I can write down the definition quickly but I want you I want to motivated and the motivation is I mean what do we do in in one variable in one variable I have let's say a rational function with a representation f X modul GX and uh let's that g of x0 is e0

um then what I look I mean EG I have some examples in the in the notes um R of X is a x modul um sorry x² x + one

then x0 equals minus1 is a point in which the the denominator vanishes so in order to to determine what what I want to call r r of x0 I I divide both the the numerator and the denominator by the the highest monomial so in this case it will be x² we get here x module x² + 1 mod x² which is of course 1 over 1 / x + 1 /

x² and and then I can substitute I mean sorry I have yeah oh I apologize um we want to evaluate at at at Infinity right so I mean I could I could I mean I could also Define it in a in a point in which the the denominator vanishes but I I I want h i mean I want to to see what we do in the one variable

case when we substitute so um I write RX is equal to um x² modle x² ided by X divided by modulus by x² + 1/ x² and then um if if the the ground field is the real numbers then I can say this is this is one divided by 1 / X+ 1 /

x² and when x ts to Infinity r 1 / X tends to 0 and 1 / x s also tends to 0 so I get 0 + 0 1 over 0 + 0 is is infinity so I would Define r at Infinity to be one over zero which is infinity now on the other hand um let's say I take the ER the rational function in in one

variable H to be X ided x² + 1 then and I want r r at Infinity I do the same thing and I mean I write RX as I divide by the the highest monomial so x / x² modul x² / x² + 1 / x² this is um 1 / x / 1 + 1 over

x² and when X tends to Infinity um the the numerator tends to zero and uh the denominator tends to one so I get r at Infinity is equal to 0 ided by 1 + 0 which is 0 yes yeah sorry madam one question um I know because last time you said that we don't really do lital we don't apply lital

because I don't know this has some property which we might not be able to follow or no no I mean we don't want to explain it ER maybe I I misspoken or maybe you misheard me but back back in the last session I think I did say ER that in in in the case of real numbers This is lupal er okay okay then I I misunderstood or I misheard yeah yeah my yeah whatever it I mean in the case of real numbers what I'm doing here I'm explaining lupal in in kind of simple terms but but of course we are

interested in the general case I mean where the field might might be just some finite field ER so but but we take this this analogy from the real numbers and we we take it as a definition this is another kind of leap of faith that things are going to sort out if we just imitate the the case of re numbers but but but is is is this something that like is proven to be correct that can do in in our particular casee or I mean remains unknown it's a definition I I I will give you a definition so the definition does not

need any proof what needs to be proven is that if you use this definition for some kind of statement I mean if you then claim that whatever some some object satisfy this definition then you need to prove it okay okay F off ER so okay so but but so how how do we extract a definition from from the setup what's going on is is I mean in the first case ER we had um a polinomial of degree two divided by

a polinomial of degree one so like in itively you could say I mean the degree two tends to Infinity faster than degree one polom so if if the the degree of denominate of the numerator is bigger than the degree of the then I would definit I would want to to say that at Infinity this rational function is infinity and the second case is the opposite it the second case I have a degree one polinomial and divided by a degree 2 polinomial so when I when I evaluate it

at Infinity it will be um I mean it will be zero right because the degree 2 polinomial tends tends to Infinity faster than the degree polinomial and what happens in the case of a a ER pols of equal degree if um R is a let's say a x² + H + 2 divided by 3 x² plus um 4

x + one now again I do the same trick I I want to I want to evaluate r at so I do this lupal trick I divide by the the the highest the highest monomial let's say x² um and here I get I get three plus 4/ x + 1/ X2 and when X tends to

Infinity H this I get H 1 +0 because and uh in the denominator I get H three 3 + 0 + 0 so this is equal to 1/3 in other words when the degrees is I mean when the degree of the numerator and the denominator is equal I simply take the the coefficient of the highest monomial in numerator and I divide it by the Coe

eent of the highest monomial in the denominator and that's that's my my answer okay so this is in in the the one variable case now okay what do we do in in a k k of E I remind you it's like a it's like quotients of pols that are both pols over e so if I

want to evaluate r at Infinity at O then I need a notion of degree and the notion of degree for pols over e is is a bit um a bit more involved than just ordinary degree of polom in in two variables namely and the degree of f for f in k okay but in K of e we have a

relation I mean in um y² is = to X Cub + a X plus b so I would want to say that the the the degree relation between the X variable and the Y variable is 2/3 and if the degree relation between the X variable and the the Y variable is 2/3

um then I might as well say that so so we want or we set the degree of the polinomial Y to be three and the degree of the polinomial X two okay and if you you set up this then indeed the degree of this polinomial is equal to the degree of this polinomial and a then we can

Define for a polom over e in canonical form let's say F FX Y is some VX + y WX H and now I have all the components I I can Define the degree of f to be um I mean how do we Define the degree of a polinomial that is a sum of

of two things it's it's the maximum of the degree of each of the summons so the degree of B h v is just a polinomial in X so I take the the ordinary degree of v as a polom in X and I multiply it by two because remember I want to consider that the degree of X is two and the other sum is a 3 plus 2 * the degree

of the the polinomial W as a a ordinary polinomial in one variable okay so this H this is a a definition of of degree that is a reasonable in in this in our context I mean because we have this uh this relation on E but now that I have the notion of degree I can I can do what I did I mean I can Define r of O just as I did in the one variable

case so for a rational function on E ER maybe I write it in in some representation it this this will not depend on the the representation um distinguish in three

cases the first case is that the degree of f is smaller than the degree G remember f and g are pols over if the degree of f is smaller than the degree of G we Define r at the point of infinity to B Infinity sorry to B

zero if on the other hand the degree of G is bigger is smaller than the degree of f maybe I write it like this degree of f is bigger than the degree of G then I Define r of O to be Infinity this this is in the projective and

um actually I have four cases not three or maybe yeah three cases but two cases here um if the degree of f is equal to the degree of G then I I need to distinguish between two cases basically if

H this degree is even if this degree is even remember by our definition of degree if the degree of a a polinomial over e is even it means that the highest monomial does not does not involve the variable y right it is a I mean it is some X some X to to the power to some power then I I said um uh then the highest monomial of

F and the highest monomial of of G are of this form H then I know what I need to do I mean then if I do this lupal procedure I divide by x to the N everything vanishes apart from the the Coe I of the the highest

monomials we Define r of O to Simply Be A over if H the degree of f which again we assume is equal to the degree of G is is odd it means that the highest monomial involves y so let's say um a x to the n y b x to the n

y are in the highest monomials of of f and g respectively and I I Define r of O To Be A over Okay so um now we have

defined I mean we have to finded a the the value of a a a rational function at the point at Infinity it could be an element in the field but it could be Infinity so what I get is that I I now have a full definition for the induced function of a rational function on e as an actual actual function on the K rational points into the projective line is this clear this is a it's a bit

tricky but uh but I think it's it is straightforward maybe we do an example this one one question regarding this last thing so to like the conclusion that isn't it a function is because you just showed that it's well defined also on the Zero Point yeah I mean because what I said earlier I mean before we had we already defined if you give me a rational

function I could could have defined it on the finite K rational points into the projective line that is I mean if the value if you take you take a point p here if R is regular at P you simply send it to r r p and if R is not regular this is the case and in The Irregular case you send it to Infinity right because P1 is K Union some additional point which we

call the point at Infinity is this is this clear or no yes thanks okay so so we already had a definition of you take a rational function you ER want to evaluate it at at any any K rational point which is not the point at infinity and you get we can I mean we can uh map it into any some point on the projective line but to to make this this picture more ER more nice we want to define the the the

russal function at the point at infinity and then we have kind of like a the the projective um I mean the projective curve or the projective points on the curve mapped into the into the projective line and the way to do it is as I defined with with the motivation from calculus lupal Rule and so on but with this this caveat that we need to to define the degree of of a polinomial over e in in a slightly more um involved way but but I guess also something that is

very natural coming from the relation that he presents okay so let's do an example I take um I guess we can say the real numbers and um rxy ah and my equation is the usual so so e is Y2 = X Cub + 8 x + B and rxy is the

following quotient X Cub + 2x + y Plus h 2 x to the 4 y divided by x + x Cub sorry X x² plus uh 5 X Y Cub this is a legitimate rational E we can write rxy equals

um X Cub + 2 X Plus y 1 + 2 x to 4 div divided by m x + plus y um 5x to the 4+ 5

that's 5 x uh yeah yeah so how can the when you define RX of Y the first time uh how can you say that it belongs to the bomal modulo the curve equation here because you have X4 and Y cubes I didn't say it is in in in canonical okay H it is I mean rational functions can come in different flavors of course

we can always transform them into canonical form but if you look at the definition that I gave I didn't assume that F that R is is is given in okay it was just any any question because I I I mean for this definition it doesn't really matter you simply look at the highest monomial and and that's that's the only thing you care about so but okay but I did but here um in the I rewrote the the function by I substitut it instead of Y Square in the denominator I

substituted of course X Cub plus a X plus b okay so I did I did a a beginning of transformation into canonical form but it's not it's not still not a um or if you wish I I see I I transformed the the the the numerator and denominator separately into canonical form but it is not the canonical form of the the rational function anyway so I can I can do this uh this substitution and now what is the degree of a of both ER both

sides here I have um in the numerator I have the m i I have two summons so it's the maximum between the degree of the first is um 2 * 3 right because the degree of the degree of of it is an ordinary polinomial is three we're talking about one the degree of this this polinomial is an ordinary polinomial is three but in in our convention the degree of of X is I mean when I consider it as as a I

mean over e the degree of X is two so this is two time three h and the other summon I have three because I have y here h plus 2 * 4 okay and this is and this is 11 yes sorry mat I I got lost here um so you said that we consider X so the first monomial if we see it as as regular polinomial this has degree three of

course so where's the two times because remember that this this goes no no sorry I didn't ask correctly so I agree with what you said in the definition but then in the other side shouldn't we have two * 4 and then times whatever y was I think three the degree of Y is three but but how some they they add yeah my bad my bad I'm stupid sorry yeah thank you no it's not stupid I did the same mistake last time so don't don't be too hard on yourself thank you h but what what do I have in the uh

um I I have 2 * 2 I mean the maximum between 2 * 2 and three times sorry 3 4 so that's also 11 okay so now if I I want to evaluate r at the point infin like again it's you can you can think of it as like I'm doing R and instead of X and Y I I substitute Infinity um but then I go to the formal definition ER I look at ER so we are in

the degree f equals degree G case and both of them are odd okay so I I look at the the highest monomial and I I simply divide the coefficients the highest monomial is in this case it will be 2 x to the 4 * Y and and below I have a 5 x to 4 y right these are the highest monomials and then R of all would be

simply 2 over five okay I know it can be a bit confusing at first but um no it makes sense it makes sense it makes sense okay yeah yeah sometimes we do Emojis to kind of say that all is good but maybe you don't see them Madam so maybe we should just reply because otherwise it might be confusing for you because we sometimes say yeah but maybe you don't see it I'll try I mean okay

if if somebody says yes I mean yeah indeed I I miss I miss the the Emojis sometimes but okay so This was um this was one step ER to tell you what we're heading to I mean when a when are I mean I take a rational function ER of course I can I can say when the rational function has a zero it means that it's a point on the curve that um so maybe I just write the

our a rational function on for point in e k H or let's say a point p is called a zero of R if R of p is equal to and a

pole our if R of p is equal to infinity and what we will be interested next few sessions is to to understand to what extent the zeros and the polls determine the rational function it it will turn out that they almost completely determine the the rational function and if they almost completely determine the rational function then I

can cook up rational functions by just finding a set of zeros and poles and you know some additional information and this will this will be a pivotal in the construction of the well pairing yes just one question here um seeing the definitions that you know give for zeros and poles this sounds similar to when you do analysis of functions and you define maximums minimums and inflection points because you also need to substitute when you derive and you substitute by zero you

might also need to run uh lital for derivating and you get first and second derivate there which kind of also helps you to Define how a function behaves uh it does that have anything to do like any kind of it's just similar by chance or there are things behind the scenes that are connected H this it's a bit I think you you are a you're a bit mixing ER the the what you what you do in calculus when you you look at the maximum and the minimum ER these are not ER zeros and poles ER or you could say they are zeros I mean the maximum and the minimum are

zeros of the derivative yeah exactly yeah that's true true but but apart from the but this is again this is typically you don't consider a a quotient so okay so zeros yeah zeros are zeros you know that's that's defined for any function um polls are only in the case of when you quotients and puls you don't usually use the the PS for ER in in ordinary calculus yeah that's true right so so it's a bit different but it it turns out

that the zeros and the poles in our case and there is I I just didn't want to get into more analysis but there is a treatment in a analysis of complex complex numbers or complex functions there is a a a lot of use of zeros and polls and this this helps you to to classify complex functions of certain kinds but this I thought it would be too much of a detour to to get into so so for us I mean we have a a natural notion of zeros and PS of a rational

function in our case and and we'll just take it from there enough make sense thank you yes Nicholas yeah sorry I think I missed the step uh can you go a little bit above yeah here you say r0 equals R infinity infinity I understand this because the degree are 11 and 11 on both on top and and bottom ah no no no no here um what I suggest to um again we have formal definitions but to H to kind of gather things quickly it's it's useful in our cases most of the

time to regard a o as having two coordinates being Infinity okay and where do does the r0 equal 25 come from again when because because because then I take the the highest monomial in the numerator I take the highest monomial in denominator yes and I divide the the leading coefficients but but why so I'm confused you you're looking at the highest monomials and you say r0 equal 25 but why was it useful to look at the degree then I'm sorry because I needed to know

I mean if the degree of the numerator is bigger than the degree of denominator then I want to say that R of of O is infinity right if because if the degree if the degree of the top is is is bigger than the degree of the bottom then then the top tends to infin faster than the bottom yeah okay Al so it's because you have the same degree on top and the bottom that you can say that now I'm looking at the monomals at the highest exactly and and I I needed I needed a notion of degree and for the other cases okay or in fact also for this case I

mean okay okay thanks all right uh Famous Last questions then uh see you next week cheers have a nice weekend cheers