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#251 2013-07-22 05:33:22

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

Re: Define the intersection points of polynomials

true i think

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#252 2013-07-22 05:41:50

bobbym
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From: Bumpkinland
Registered: 2009-04-12
Posts: 84,605

Re: Define the intersection points of polynomials

Here is another one:

We get x = 5, which corresponds to solving that equation over GF(19)


In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

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#253 2013-07-22 05:43:10

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

Re: Define the intersection points of polynomials

I verified it with the calculator. It seems right!

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#254 2013-07-22 05:43:59

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 84,605

Re: Define the intersection points of polynomials

I changed post #252, so please check again.


In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

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#255 2013-07-22 05:47:31

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

Re: Define the intersection points of polynomials

It is 135=2mod19 which is right. (Always I think..)

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#256 2013-07-22 05:57:35

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 84,605

Re: Define the intersection points of polynomials

Okay, then for the sake of argument because we have nothing else to go on let's assume Mathematica can solve your equations over the GF you want. This is a pragmatic decision, if we assume it can not we are done right here. Let's proceed with the assumption and see what the heck happens.


In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

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#257 2013-07-22 05:58:14

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

Re: Define the intersection points of polynomials

Ok!

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#258 2013-07-22 06:00:18

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 84,605

Re: Define the intersection points of polynomials

Now, I can continue to ask the remaining questions.

GF(p) what is p going to be? I would very much like to have a p smaller than 2^128. Is this possible?


In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

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#259 2013-07-22 06:02:15

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

Re: Define the intersection points of polynomials

Yes try  a small prime number. I think if it is solvable for small p it will be solavble for larger p's..pick 113 i think it is prime

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#260 2013-07-22 06:05:56

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 84,605

Re: Define the intersection points of polynomials

This is what we agree on:

So the leading coefficients of each of the 4 given polynomials will be ∈ GF. The points given will have x's and y's that are ∈ GF. Now you are requiring the points of intersection of the 4 quadratics to also be ∈ GF?

A Gf(113) will be used.

Agreed?


In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

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#261 2013-07-22 06:09:23

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

Re: Define the intersection points of polynomials

Yes. I want to know if we can find the intersection points of the polynomial as syou didi wi th real numbers

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#262 2013-07-22 06:11:13

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 84,605

Re: Define the intersection points of polynomials

I understand the problem well enough to work on it. I am currently a little busy with 4 other problems and will start modifying the cubic code that I used to solve the other problems. This will take some time.


In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

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#263 2013-07-22 06:16:55

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

Re: Define the intersection points of polynomials

Ok!!!

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#264 2013-07-22 07:51:51

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 84,605

Re: Define the intersection points of polynomials

So far I have found it very difficult to even construct 4 quadratics and 2 points of intersection all of them over Z let alone GF(113).


In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

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#265 2013-07-22 07:53:11

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

Re: Define the intersection points of polynomials

why?

Last edited by Herc11 (2013-07-22 07:54:45)

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#266 2013-07-22 07:55:39

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 84,605

Re: Define the intersection points of polynomials

It is a little early to say.  But it does suggest they are rare. That means that getting an answer to them in the required form is very rare.


In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

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#267 2013-07-22 07:57:25

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

Re: Define the intersection points of polynomials

If the set is small maybe is a good choce to use a bigger p.

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#268 2013-07-22 08:03:07

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

Re: Define the intersection points of polynomials

You can use Newton Interpolation i.e the formulas for a that I have posted eralier to compute the polynomials.

What I mean is that you can preselect the common x_0 y_0 x_1 y_1 for the four polynomials. Then You can use the postedd formulas for recovering a_0 a_1. Then for each polynomial you can select a different x_3 y_3. Then by the posted formula again you can compute a_3. 

After that tou will have 4 polynomials and the set of equations is ready.

The problem is to "pretend" that you do not know the 2 intersection points x_0x_1 y_0 y_1 and try to solve the previous system.

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#269 2013-07-22 08:09:53

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 84,605

Re: Define the intersection points of polynomials

I can do all that. The trouble is the answers are not coming back as integers let alone 1 - 113.


In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

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#270 2013-07-22 08:14:08

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

Re: Define the intersection points of polynomials

What do you mean?They have to return as elements. If not something is wrong...

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#271 2013-07-22 08:24:11

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 84,605

Re: Define the intersection points of polynomials

I think I have a method now that is working to generate the problems. I will go back to work on the solutions.

Half way through with it. I need a little sleep, see you later hopefully with a full solution.


In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

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#272 2013-07-22 21:16:29

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 84,605

Re: Define the intersection points of polynomials

Hi;

I have 4 parabolas that  intersect at two points. The leading coefficients are 5, 22 17, 29. I have 4 points on those 4 parabolas, 1 on each parabola.

They are:

(3,13), (3,47), (3,37), (3,61)

You can see that so far all the conditions have been met. Now I must calculate from this information the points of intersection modulo 112.

The first idea to solve it failed completely. I think I see what went wrong.


In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

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#273 2013-07-22 22:01:24

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

Re: Define the intersection points of polynomials

What went wrong?? mod113

Last edited by Herc11 (2013-07-22 22:02:03)

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#274 2013-07-22 22:10:52

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 84,605

Re: Define the intersection points of polynomials

That will be a problem when I get to that.

What went wrong first is the choice of points. You notice the x's are all 3. This caused the system to not have a unique solution. Why? I do not know! I will have to adjust the test problem to use all different or at least some different x's.


In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

Offline

#275 2013-07-22 22:14:30

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

Re: Define the intersection points of polynomials

The choice of same x's would a problem if wou were solving the problem over real numbers?

Dιδ you used x=3 for creating the parabolas?

How did you construct the parabolas?Can you be sure that they have 2 intersection points? Are the intersection points known?

Last edited by Herc11 (2013-07-22 22:19:53)

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