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#151 2013-06-21 08:31:23

Herc11
Member
Registered: 2013-06-19
Posts: 169

Re: Define the intersection points of polynomials

I think it has not to do with interpolation. See in #146 post

the formula for computing a_3

if the unknowns are x0 y0 x1 y1 x2 y2

and known the x3 y3 and a3

I think that by having a set of 6 equations you can define x0-y2.

Also, bobbym

posted another one problem

Last edited by Herc11 (2013-06-21 08:34:00)

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#152 2013-06-21 08:33:49

bobbym
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From: Bumpkinland
Registered: 2009-04-12
Posts: 87,258

Re: Define the intersection points of polynomials

Unless you eliminate the a0, a1 and a2 you will have more than 6 variables. You will have 9 variables. This means 9 cubics.

Fortunately I have a way around that.


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

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#153 2013-06-21 08:35:38

Herc11
Member
Registered: 2013-06-19
Posts: 169

Re: Define the intersection points of polynomials

Why to eliminate?

Simply substitute the a0 a1 and a2 with x0-y2.

See post #146.

Dont use a0..etc if it confuses you...

You have only to solve for x0..y2.

And six equations like that in #146 are enough

Last edited by Herc11 (2013-06-21 08:43:29)

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#154 2013-06-21 08:42:27

bobbym
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From: Bumpkinland
Registered: 2009-04-12
Posts: 87,258

Re: Define the intersection points of polynomials

Aren't the a0, a1, a2, a3 the coefficients of the cubic?


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

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#155 2013-06-21 08:46:48

Herc11
Member
Registered: 2013-06-19
Posts: 169

Re: Define the intersection points of polynomials

Yes they are coeffients.

But see equation in #146.

Then suppose you have 6 equations like #146.

You know a3 y3 x3, and you dont know x0x1x2 y0y1y2.

By a set of 6 equations isn t the set of equations solvable?

Our aim is to define the intersection points i.e.  x0x1x2 y0y1y2.

Once you solve the set of the 6 equations you will have discovered the  x0x1x2 y0y1y2.

If you d like you can then define a0=y0 a1=... and  a2=...

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#156 2013-06-21 08:55:20

bobbym
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From: Bumpkinland
Registered: 2009-04-12
Posts: 87,258

Re: Define the intersection points of polynomials

Hi;

Yes, I see that. I am agreeing with you. They are replaced as you did in post #146, so did I. Eventually all of them are replaced and just the 6 variables we want remain.

I did not understand you were saying the same thing I was.


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

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#157 2013-06-21 08:58:53

Herc11
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Registered: 2013-06-19
Posts: 169

Re: Define the intersection points of polynomials

Ok:)

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#158 2013-06-21 09:07:35

bobbym
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From: Bumpkinland
Registered: 2009-04-12
Posts: 87,258

Re: Define the intersection points of polynomials

Hi;

I am currently busy answering a question. I will need some sleep right after. I will work out the problem I posed and get the solution as soon as I wake up. I am really beat.


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

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#159 2013-06-21 09:10:19

Herc11
Member
Registered: 2013-06-19
Posts: 169

Re: Define the intersection points of polynomials

ok ok.

I think that anonimnystefy can solve it as he solved the quadratics...

Last edited by Herc11 (2013-06-21 09:17:08)

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#160 2013-06-21 09:23:52

bobbym
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From: Bumpkinland
Registered: 2009-04-12
Posts: 87,258

Re: Define the intersection points of polynomials

He is sleeping by now, I think. I need some rest. see you later and thanks for the problems.


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

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#161 2013-06-21 09:29:11

Herc11
Member
Registered: 2013-06-19
Posts: 169

Re: Define the intersection points of polynomials

Thans both of you!

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#162 2013-06-21 09:31:24

Herc11
Member
Registered: 2013-06-19
Posts: 169

Re: Define the intersection points of polynomials

By the way are you students or college students?

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#163 2013-06-21 10:31:46

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 15,544

Re: Define the intersection points of polynomials

Hi guys

I haven't had time to post here. Have you got the interpolation thing to work?


“Here lies the reader who will never open this book. He is forever dead.

“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

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#164 2013-06-21 17:28:09

Herc11
Member
Registered: 2013-06-19
Posts: 169

Re: Define the intersection points of polynomials

Hi anonimnystefy,

See the posts #133,135 and 146,147.

The aim is to solve a set of 6 equations of a_3 with the information provided in post 133 and 135.

I thinh that you can with mathematica.

You have 6 equations, with 6 unknown variables. I cant see why this is not sovlable.

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#165 2013-06-21 19:23:05

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 87,258

Re: Define the intersection points of polynomials

Hi;

I have solved the problem given above using just 6 equations.


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

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#166 2013-06-21 19:27:18

Herc11
Member
Registered: 2013-06-19
Posts: 169

Re: Define the intersection points of polynomials

Thats nice!
So I think one can say that If you have n intesection points, and someone provides you with the lead coeficient and a point of 2n polynomials you can define the intersection point.
Scool student?

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#167 2013-06-21 19:37:05

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 87,258

Re: Define the intersection points of polynomials

Hi;

Yes, I would say so but the actually calculation becomes more and more difficult as n increases.

I am not a student.


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

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#168 2013-06-21 19:41:05

Herc11
Member
Registered: 2013-06-19
Posts: 169

Re: Define the intersection points of polynomials

What do you mean "difficult"?

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#169 2013-06-21 19:49:46

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 87,258

Re: Define the intersection points of polynomials

Numerical difficulties. Although A,B and C in my problem are integers the equations have enough instability in them to run off to the complex plane. Already Geogebra cannot create the points accurately enough and it has 16 digits of precision.


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

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#170 2013-06-21 19:52:49

Herc11
Member
Registered: 2013-06-19
Posts: 169

Re: Define the intersection points of polynomials

Ok, if the operations were over a galois field will have  any difference?

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#171 2013-06-21 19:57:20

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 87,258

Re: Define the intersection points of polynomials

You mean a set consisting of {1,2,3,4,...p} where p is primes? If so, that is what I did.

You noticed the ai's I provided for the problem? They are floating point numbers. Geogebra lost 4 digits in computing them. That suggests a condition number for the problem of around 10000.


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

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#172 2013-06-21 20:00:09

Herc11
Member
Registered: 2013-06-19
Posts: 169

Re: Define the intersection points of polynomials

Something like that...

The problem is that when you work over Finite fields,

addition multiplication and division are not equal to standard operations. operations are executed modulo the prime number etc..etc..

Last edited by Herc11 (2013-06-21 20:06:32)

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#173 2013-06-21 20:08:15

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 87,258

Re: Define the intersection points of polynomials

Yes, I know my grasp of theory is horrendous.


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

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#174 2013-06-21 20:16:22

Herc11
Member
Registered: 2013-06-19
Posts: 169

Re: Define the intersection points of polynomials

I am familiar with Galois Fields and the operations executed over GFs but it is not ot easy to explain...

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#175 2013-06-21 20:33:52

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 87,258

Re: Define the intersection points of polynomials

A small thing to improve performance of solving that would be to have the coefficient of the constant term rather than the leading coefficient.


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

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