Advice for Research Mathematicians

Spread polynomials arise in Rational Trigonometry as variants of the Chebyshev polynomials of the first kind. However the spread polynomials arise in a purely algebraic setting, without any need for appeal to "transcendental functions" which can't actually be evaluated -- such as cos x or sin x.

However they capture much of the same features as do the more traditional Chebyshev polynomials -- with one big difference: there is a remarkable factorization, discovered by Shuxiang Goh, of the spread polynomials which is not shared by the Chebyshev polynomials. From a number theoretical perspective, this establishes the primacy of the spread polynomials -- and many new areas of investigation, suitable also for amateur researchers are then opened up about the nature of these factors.

There are quite a few more detailed formulas in this lecture in the direction of "special functions" or "orthogonal polynomials" -- some of them obtained by M. Hirschhorn (UNSW) and S. Goh (UNSW). There is a lot to digest here.

This is part of a series of videos where we give general and specific advice to prospective researchers in mathematics -- from the amateur to the professional level. In particular we are interested in identifying fruitful areas of study that even amateurs can, with some application, make contributions to.

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Advice for Mathematics Exploration (for Members only): https://www.youtube.com/playlist?list...

Solving Polynomial Equations (Members only): https://www.youtube.com/playlist?list...

Hexagrammum Mysticum (Members only): https://www.youtube.com/playlist?list...

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Maxel Inverses and Orthogonal Polynomials (for Members only): https://www.youtube.com/playlist?list...

Triangle Centres: https://www.youtube.com/watch?v=iLBGX...

Six: An elementary course in pure mathematics: https://www.youtube.com/playlist?list...

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Universal Hyperbolic Geometry: https://www.youtube.com/playlist?list...

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0:00

Hello everyone i'm Norman Wildberger and thanks for joining me i've given you some advice

0:04

with respect to research mathematics in the direction of rational trigonometry and spread

0:11

polynomials so this is a continuation from my last lecture okay so we've introduced the basics

0:17

of rational trigonometry and now i want to fast forward to the section on the spread polynomials

0:26

so we've just finished the proof of the main laws of rational trigonometry and now we can talk about

0:33

the spread polynomials and how they arise naturally so a good place to start is with

0:38

this result which i call the equal spreads theorem which explains what happens if you have

0:44

three lines say l1 l2 and l3 so that the spread between l1 and l2 equals the spread between

0:52

l2 and l3 so sort of like three lines okay emanating from some point and the spread

0:58

between these two lines okay equals the spread between these two lines let's call that common

1:03

spread s and what we're interested in is the spread between the first line and the third line

1:09

what are the possibilities for that okay now in terms of a diagram here are sort of two

1:16

possibilities for that one ways of thinking about it okay so in this case here they have

1:21

a sort of a common intersection here and here's the spread of s between l1 and l2 and here's the

1:25

spread of s between l2 and l3 and i remark that where we put this little segment um

1:31

is is flexible okay so there's two lines here and there's actually sort of four uh sectors

1:37

that they determine and we could put this little line segment in any one of those four sectors

1:43

so we could also describe this this spread here it's also going from here to uh to here

1:52

and or or from here to here or from here to here okay so these two coven spreads

1:57

s and then between l1 and l3 that spread between these two lines is described by this

2:04

new spread r and we're interested in the relationship between r and s

2:07

now here's another sort of configuration that still sort of satisfies that same criterion so

2:12

here's l1 and here's l2 and they make a spread of s here and l2 and l3 make a spread of s

2:18

and then the spread between l1 and l3 is r okay and the statement is that then

2:27

this spread between l 1 and l 3 is either 0 or it's equal to 4 times s times one minus s

2:38

okay so to prove that so suppose this spread s of l one l three is r okay then from the triple

2:46

spread formula which tells us the relationship formed by three lines spreads formed by three

2:52

lines okay so maybe it's uh easier to think about this here's this more obvious triangle

2:58

but in fact it works in either case so there are spreads of s s and r so the triple spread

3:04

formula will tell us then that the sum of the spreads all squared so that's 2s plus r l squared

3:12

equals 2 times the sum of the squares of the spreads so s squared plus s squared plus r squared

3:18

or 2s squared plus r squared plus this additional cubic term four times the product of the spreads

3:25

which will be s squared r now if you bring everything to one side and sort of simplify things

3:30

then you'll see that this thing factors as r times r minus 4s times 1 minus s and the reason it

3:38

factors is because the solution r equals 0 is kind of obvious that is one possibility in this diagram

3:46

is for l1 and l3 to actually coincide say this diagram here right if we sort of move l3 so

3:52

it's coinciding with l1 then that will satisfy the conditions in that case r is equal to zero

3:58

so that's sort of a an automatic solution and then since it's a quadratic equation in r the

4:04

other solution is obtainable immediately just by a linear equation so we get r equals 0 or r equals 4

4:11

times s times 1 minus s now this particular function is called the logistic map and it

4:17

plays an important role in chaos theory here's the graph of it's an inverted parabola from 0 to 1 it

4:24

goes from 0 to 1. so you can regard this if you like as a map from the interval zero one to itself

4:32

but it's not a one to one map it because well as you can see you know any most values there's two

4:39

um places where those values are taken on so in some sense this what what it does is sort of

4:44

folds the interval over onto itself in a two to one fashion and this is a quite an interesting and

4:50

important example in chaos theory so it's a rising here this is actually the second spread polynomial

4:59

okay so more generally well first of all we call this s2 of s so that's four times s times one

5:06

minus s so that's logistic map so then the spread polynomial s sub n of s may be defined recursively

5:13

by setting s 0 of s to be 0 and s 1 of s to be s and then here is the inductive recursive rule s

5:22

n of s equals two times one minus two s times s n minus s minus s n minus two of s plus two s so it

5:33

shares some of the features of something like the Chebyshev polynomials of the first or second kind

5:38

but this plus 2s is a little bit different it makes it a little bit different and

5:45

and then we hear some graphs of a few of them so this would be s3 and here's s4 and here's a

5:55

bunch of them okay and you can see that in the interval 0 1 maps interval 0 1 to 0 1 they'll

6:02

bounce back and forth in this in this box in a very pleasant way which we talked about before

6:10

now the the connection with the triple spread formula is the following that this recursive

6:19

definition of the spread polynomials that we've defined has the property that if you take s n

6:26

minus 1 and s and s n of s now those three numbers for any s satisfy the triple spread formula

6:36

so this is uh an algebraic reflection of the fact that what this thing is is measuring is

6:43

you so imagine you have say two lines okay making a spread of s and then imagine sort of reflecting

6:49

this first line in the second line so now we have a third line okay that's s and that's s okay and

6:55

then we can keep going you know we can uh reflect this one in this one to get a fourth one and this

6:59

is s and then this is s and then this is s and what we're doing is we're then calculating the

7:05

spread between the first and the last line there's in this case there's three equal spreads okay

7:10

so there's going to be the connection between the s and this final spread is that it's

7:18

s sub three of s okay and in general what happens is that if you know s of n minus one then you know

7:25

that spread and then you add one more s so then you get s n now there's a number of different

7:31

sort of interpretations or you know additional formulas for snms it's a very rich kind of thing

7:37

and here in this first line we connect with the Chebyshev polynomials t sub n of the first kind

7:44

okay i think i've told you about this one here that it can be expressed as 1 minus t sub n of

7:48

1 minus 2s over 2. so it's an affine shift it can also be expressed as 1 minus t sub n squared of

7:57

square root of one minus s and so um this thing here there seems to be an irrationality here

8:06

there's no irrationality okay here's another relation let's not talk about

8:11

that one um and here's an explicit formula uh obtained by Susheen Goh and Mike Hirschhorn

8:19

a former student of mine and a colleague of mine who have also looked at these things and they they

8:25

found this formula which is expressing s n of s as a polynomial in in powers of s with certain

8:32

coefficients which are quite attractive some binomial coefficient there and an n over n minus k

8:40

and then we also have a generating function which i've also mentioned to you found also by Mike

8:45

Hirschhorn so the generating function the ordinary generating function for s n of s if we write it

8:50

this way here is s times x times 1 plus x all over one minus x cubed plus four s x times one minus x

8:58

and uh another formula uh also found by mike here short is um the following that uh s n of

9:05

s can be written as a sort of a binomial expansion involving one powers of one minus s and s but with

9:13

these um sort of odd uh binomial coefficients and choose one and choose three and choose five etc

9:24

okay so you know uh there's probably a lot more um formulas to be found about these things i i

9:29

think these spread polynomials are absolutely really crucially important and we're going to

9:35

see a little bit more of that okay so let's go to our next slide here we are seeing what they

9:40

actually look like um in terms of some lists of them okay so here's sort of unfactored forms

9:47

s 0 is 0 s1 is s s2 we've already seen that's 4s minus 4s squared that's the logistic map

9:54

s3 is 9s minus 24s squared plus 16s cubed which factors as s times 3 minus 4s all squared

10:04

that's actually also an important one and here's s4 in expanded form and in factored form

10:11

s5 in expanded form factored form s6 also and s7 to give you a sense of things

10:21

and i've already told you about this but let me just reiterate okay that the the really

10:28

crucial property that these spread polynomials have that the Chebyshev polynomials don't is that

10:34

they have a very strong factorization so this is an observation of Susheengo when he was an honor

10:42

student with me and the observation is that s n of s factors as a product of k dividing n so overall

10:50

factors of n of some auxiliary polynomials phi sub k of s these auxiliary polynomials

10:59

spread psychoatomic polynomials or maybe just go polynomials um phi sub 1 phi sub 2 phi sub 3

11:07

5 sub 4 here some values of them phi sub 5 y sub 6 y sub 7 and phi sub 8. and

11:14

you're you're gonna be hard-pressed to find some kind of easy regular rule to describe

11:20

what's happening with these things so these are playing some kind of role

11:26

something like you know like prime numbers or something but but you know in a different kind

11:31

of way but um so for example s sub 12 uh factors in as a product of phi sub one phi sub two phi

11:39

sub three sub pi sub four five sub 6 and phi sub 12. phi sub 12 happens to be this thing here and

11:45

then you get this this expansion here which i think i've already shown you at some earlier point

11:51

so this is a much stronger factorization than the Chebyshev polynomials

11:55

enjoy and this is i think really showing clearly that sort of number theoretically

12:02

these these spread polynomials are really uh sort of at center stage and because the spread

12:09

polynomials and the Chebyshev polynomials are only connected by an affine shift so it's just really a

12:15

question of sort of you know scaling them a little bit and moving them so properties of the Chebyshev

12:21

polynomials having to do with integration and and extrapolation and things like that can easily

12:28

be restated in terms of the spread polynomials okay but it's this factorization property that

12:33

makes them particularly interesting and really brings out what i think is the number theoretical

12:39

context of things so studying these five sub sub k's here i think is a is a a massive task

12:47

that will keep people going for for generations in my view and you can be amongst the pioneers.