Define the intersection points of polynomials

Herc11 · 2013-06-21 21:43:24

Maybe its the same thing. I am not sure.

bobbym · 2013-06-21 22:43:31

I could also write the routines to generate a problem in Mathematica with exact arithmetic. Leaving Geogebra out. The error would diminish greatly.

Did you have a fixed n you needed to be solved or were you just seeing what could be done?

Herc11 · 2013-06-22 05:13:29

No, only I want to see...just being curious!

bobbym · 2013-06-22 05:22:58

Hi;

Okay, I understand.

Herc11 · 2013-06-22 20:03:03

To be honest, I want to know if by having 2n equations

(of the previous form==leading coef.+ 1 point of the polynomial)

I can define the intersection point of the n degree polynomials.

You and anomnimistefy prove that for the case of n=2, n=3 the previous case stands.

I think it stands for every n.

bobbym · 2013-06-22 21:25:17

Each set of polynomials of n degree can only intersect at n points. Each point will have 2 variables x and y. You will need 2n equations to solve for them.

anonimnystefy · 2013-06-22 22:31:46

I thought we already got to this conclusion before.

bobbym · 2013-06-23 00:13:29

The evidence supports this but there are practical considerations.

Herc11 · 2013-07-19 11:33:10

Hey bobby m,
Do you know if the set of equations that were talking about can be solved over Galois Fields?Or where to search?

bobbym · 2013-07-19 17:32:34

If you mean solved modulo 1...n, then Mathematica might be able to handle the job.

Herc11 · 2013-07-19 19:44:52

Yes I meant a Finite feild generated by a prime n or an irreducible polynomial.

I am not sure that is solvable if the equations of the set are nonlinear i.e. te polynomials are of 2-degree 3- etc...

Last edited by Herc11 (2013-07-19 19:55:22)

bobbym · 2013-07-19 21:27:11

Hi;

Do you have a worked example?

Herc11 · 2013-07-19 21:52:46

No..It is not easy to test it. I have written in C a galois field multiplier and divider for an irreducilbe polynomial of degree 128.

The addition is simply an xor.

The equations are the same that I was posted in previous posts.

I google it and the results that wre obtained confused me. e.g. I tried to read the following thesis but with no result.

If i undersstood correctly the problem is not easily solvable but I am not sure that this thesis copes the same problem as mine

https://openaccess.leidenuniv.nl/bitstream/handle/1887/4392/Thesis.pdf?sequence=1

Last edited by Herc11 (2013-07-19 21:56:23)

bobbym · 2013-07-19 23:38:35

Hi;

Reading the pdf now.

Herc11 · 2013-07-19 23:39:53

I think that when the set of equations is constituted by polynomials of degree-n, an unknown

will come up.

I dont know how

can be computed over GFs.

bobbym · 2013-07-19 23:46:16

Isn't that covered in his "power extraction algorithm?" Of course, he has gone on and on with his existence proofs but I do not see an implementation so far.

Herc11 · 2013-07-19 23:51:36

Whats this?I ve never heard it. Can this solve the aforementioned problem?

You give me hope!

Last edited by Herc11 (2013-07-20 00:32:35)

bobbym · 2013-07-20 05:04:47

He mentions it in the pdf you provided.

You should not have hope yet. Highbrows like him rarely explain anything. The chance that he will provide an example that a lowbrow like me can follow is 1 in 10000.

Herc11 · 2013-07-20 19:31:07

If we try a different approach and instead replacing the

to the formula we write

then the formula will be e.g for 3 degree polynomials

So If again we know

we need a ste of 2x3=6 such polynomials to solve the system and the system remains linear i.e. to get the afterwards that we have defined

we can solve

and recover the missing intersection points.

So it seems that it can be solved over GF. I think...

bobbym · 2013-07-20 21:14:27

Wouldn't a0, a1, a2,... all have to be members of the GF?

Herc11 · 2013-07-20 21:30:56

Yes all are members of the GF.

bobbym · 2013-07-20 21:47:51

Hi;

How can you be sure?

Herc11 · 2013-07-20 21:55:16

About what?

bobbym · 2013-07-20 22:35:08

That the a0, a1, a2... will all be integers let alone mebers of that set?

Herc11 · 2013-07-20 22:43:27

All the variables known and unknown will be elements of a Galois Field. Multiplication Division, addition(=substraction) will be defined over the GF. The result of the operations will be GFs too. They are not integers but elements of the GF. I mean that alla the problem will be defined and solved over the GF.

Math Is Fun Forum

#176 2013-06-21 21:43:24

Re: Define the intersection points of polynomials

#177 2013-06-21 22:43:31

Re: Define the intersection points of polynomials

#178 2013-06-22 05:13:29

Re: Define the intersection points of polynomials

#179 2013-06-22 05:22:58

Re: Define the intersection points of polynomials

#180 2013-06-22 20:03:03

Re: Define the intersection points of polynomials

#181 2013-06-22 21:25:17

Re: Define the intersection points of polynomials

#182 2013-06-22 22:31:46

Re: Define the intersection points of polynomials

#183 2013-06-23 00:13:29

Re: Define the intersection points of polynomials

#184 2013-07-19 11:33:10

Re: Define the intersection points of polynomials

#185 2013-07-19 17:32:34

Re: Define the intersection points of polynomials

#186 2013-07-19 19:44:52

Re: Define the intersection points of polynomials

#187 2013-07-19 21:27:11

Re: Define the intersection points of polynomials

#188 2013-07-19 21:52:46

Re: Define the intersection points of polynomials

#189 2013-07-19 23:38:35

Re: Define the intersection points of polynomials

#190 2013-07-19 23:39:53

Re: Define the intersection points of polynomials

#191 2013-07-19 23:46:16

Re: Define the intersection points of polynomials

#192 2013-07-19 23:51:36

Re: Define the intersection points of polynomials

#193 2013-07-20 05:04:47

Re: Define the intersection points of polynomials

#194 2013-07-20 19:31:07

Re: Define the intersection points of polynomials

#195 2013-07-20 21:14:27

Re: Define the intersection points of polynomials

#196 2013-07-20 21:30:56

Re: Define the intersection points of polynomials

#197 2013-07-20 21:47:51

Re: Define the intersection points of polynomials

#198 2013-07-20 21:55:16

Re: Define the intersection points of polynomials

#199 2013-07-20 22:35:08

Re: Define the intersection points of polynomials

#200 2013-07-20 22:43:27

Re: Define the intersection points of polynomials

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