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You are not logged in. #201 20130721 21:21:14
Re: Define the intersection points of polynomialsHi; 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. #202 20130721 22:02:28#203 20130721 22:06:29
Re: Define the intersection points of polynomialsYes, I know bit to compute one we would have to assign a value to p. 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. #204 20130721 22:15:14
Re: Define the intersection points of polynomialsHmmm ok. I used an extension field GF 2^128 I used a polynomial basis representation and a pentanomial irreducible polynomial for the generation of the field/. But I think that if the system can be solved on GF(p) will be solved at GF(2^128) . #205 20130721 22:23:15
Re: Define the intersection points of polynomialsNow let me get our definitions synchronized. When they say solve over the Reals for an equation they mean the roots ∈ R. When they say over the rationals that means the roots are ∈ Q. For the integers, the roots are ∈ Z. 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. #206 20130721 22:26:28#207 20130721 22:32:30
Re: Define the intersection points of polynomialsRemember what you were solving for, the a0,a1,a2... these are the roots of the equations we set up. The way I understand it the roots are the quantities that have to be ∈ GF(17). That means the a0,a1,a2... ∈ GF(17) 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. #208 20130721 22:34:59#209 20130721 22:43:17
Re: Define the intersection points of polynomialsThat is what I mean all the variables that we solve for have to be ∈ GF(17) 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. #210 20130721 22:48:09#211 20130721 22:50:21
Re: Define the intersection points of polynomialsMy feeling is that no matter how large you pick p to be it is unlikely that the variables will be elements of it. 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. #212 20130721 22:55:40
Re: Define the intersection points of polynomialsMultiplications addtion and division over the gf is not as standard opeerations. More generally all the operations executed over GF are performed modulop. e.g 5+15=20modp=3mod17. #213 20130721 22:58:03
Re: Define the intersection points of polynomialsWhat about if the equations have no solutions? The restriction of modulo arithmetic might mean no solutions. 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. #214 20130721 23:00:27#215 20130721 23:02:04
Re: Define the intersection points of polynomialsWe will only need to find one example of when there are no solutions. 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. #216 20130721 23:05:00#217 20130721 23:08:40
Re: Define the intersection points of polynomialsThere are limits to what can be computed. The only thing that is required is that Mathematica be able to solve modulo p  1. I will begin to investigate what it can do. 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. #218 20130721 23:10:15#219 20130721 23:13:25
Re: Define the intersection points of polynomialsIt can solve lots of equations using modular arithmetic. 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. #220 20130722 02:17:39#221 20130722 12:58:31
Re: Define the intersection points of polynomialsNot yet. I am going to have to find my notes on the original problem. I have forgotten everything about it. I will post immediately when I have something. 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. #222 20130722 16:32:54#223 20130722 16:41:49
Re: Define the intersection points of polynomialsStill putting together the notes of the last problem. Once that is done I will have some idea whether or not I can do what you ask. 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. #224 20130722 16:52:05#225 20130722 20:19:31
Re: Define the intersection points of polynomialsI hope not. Hopefully, it will just do what is required but even if it does it will take some time to find a counterexample. 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. 