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Topic: The actual math, advanced polynomial factorization
Replies: 14   Last Post: Sep 29, 2004 9:43 AM

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 Andrzej Kolowski Posts: 353 Registered: 12/6/04
Re: The actual math, advanced polynomial factorization
Posted: Sep 27, 2004 8:08 PM

jstevh@msn.com (James Harris) wrote in message news:&lt;3c65f87.0409270231.71cdcf01@posting.google.com&gt;...
&gt; David Kastrup &lt;dak@gnu.org&gt; wrote in message news:&lt;x51xgp8b3o.fsf@lola.goethe.zz&gt;...
&gt; &gt; jstevh@msn.com (James Harris) writes:
&gt; &gt;
&gt; &gt; &gt; David Kastrup &lt;dak@gnu.org&gt; wrote in message news:&lt;x53c16xb8r.fsf@lola.goethe.zz&gt;...
&gt; &gt; &gt;&gt; jstevh@msn.com (James Harris) writes:
&gt; &gt; &gt;&gt;
&gt; &gt; &gt;&gt; &gt; The following factorization is useful:
&gt; &gt; &gt;&gt;
&gt; &gt; &gt;&gt; For what?
&gt; &gt; &gt;&gt;
&gt; &gt; &gt;&gt; &gt; f^2((m^3 f^4 - 3m^2 f^2 + 3m) x^3 - 3(-1+mf^2 )x u^2 + u^3 f) =
&gt; &gt; &gt;&gt; &gt;
&gt; &gt; &gt;&gt; &gt; (a_1 x + uf)(a_2 x + uf)(a_3 x + uf)
&gt; &gt; &gt;&gt; &gt;
&gt; &gt; &gt;&gt; &gt; Note that x, m, f, and u are all independent variables.
&gt; &gt; &gt;&gt;
&gt; &gt; &gt;&gt; No, they aren't independent. They are related by the equation
&gt; &gt; &gt;&gt;
&gt; &gt; &gt;&gt; f^2((m^3 f^4 - 3m^2 f^2 + 3m) x^3 - 3(-1+mf^2 )x u^2 + u^3 f) =
&gt; &gt; &gt;&gt; (a_1 x + uf)(a_2 x + uf)(a_3 x + uf)
&gt; &gt; &gt;&gt;
&gt; &gt; &gt;
&gt; &gt; &gt; No. They are not. Notice that you can give any value you wish for x,
&gt; &gt; &gt; m, f and u without checking each variable against the other because
&gt; &gt; &gt; they are not constrained against each other.
&gt; &gt; &gt;
&gt; &gt; &gt;
&gt; &gt; &gt; But notice that you can't just give a value for a_1, a_2 and a_3
&gt; &gt; &gt; because they ARE constrained by the values of x, m, f and u.
&gt;
&gt;
&gt; That's the mathematics, but notice what I face, which is what I've
&gt; faced from sci.math posters for YEARS, in the reply by David Kastrup.
&gt;
&gt; &gt;
&gt; &gt; That's nothing to "notice", that is what you may want to define.
&gt; &gt; Prima facie, the above equation just expresses a relation between all
&gt; &gt; of the variables and does not differentiate anything among them. In
&gt; &gt; the context of defining a particular task associated with the
&gt; &gt; equation, you can assign meaning to various variables. You always
&gt; &gt; muddy up things wildly. In particular, for the purpose you want, you
&gt; &gt; _choose_ the following characterizations:
&gt; &gt;
&gt; &gt; a) x is a symbolic variable used for defining a polynomial. It does
&gt; &gt; not assume any value at all.
&gt; &gt; b) m, f and u are _declared_ to be free variables: you try making a
&gt; &gt; statement that will hold for all choices of them.
&gt; &gt; c) a_1, a_2 and a_3 are then (though not necessarily uniquely)
&gt; &gt; determined by that choice and [are in the coefficient space for the
&gt; &gt; polynomial defined over x] Correction in superseded post: this is
&gt; &gt; of course not the case when the polynomial can't be factored into
&gt; &gt; linear terms in the given coefficient space, like when factoring
&gt; &gt; the real polynomial x^2+1.
&gt; &gt;
&gt; &gt; In order to make something like this make sense, you have to declare
&gt; &gt; _how_ m, f and u are supposed to be chosen: what possible values they
&gt; &gt; may assume. In addition, you have to declare the ring or other
&gt; &gt; algebraic structure in which a_1, a_2 and a_3 are supposed to be.
&gt; &gt; Only then can you start making meaningful statements.
&gt; &gt;
&gt; &gt; And in fact, you have repeatedly changed your mind about those basic
&gt; &gt; premises you have not bothered to fix in APF, and are lying about
&gt; &gt; this when defending it.
&gt;
&gt;
&gt; f^2((m^3 f^4 - 3m^2 f^2 + 3m) x^3 - 3(-1+mf^2 )x u^2 + u^3 f) =
&gt;
&gt; (a_1 x + uf)(a_2 x + uf)(a_3 x + uf)
&gt;
&gt; and f, m, x and u are independent variables, while a_1, a_2, and a_3
&gt; are dependent variables.
&gt;
&gt; Such a simple thing, but notice all the verbiage from David Kastrup.
&gt;

Yes, f, m, x, and u are independent and a_1, a_2, and a_3 are not.
But the statement you are discussing, i.e., divisibility of
a_1, a_2 or a_3 by f in the algebraic integers is not really a
statement about f. It is a statement about a_1, a_2, and a_3,
conditional on f. That is, the statement itself is about
*dependent* variables. So any vague principle you have in
is a bogus argument based on a nonmathematical "principle"
that you think applies in this case. It does not.

Andrzej

&gt; It's not about what's mathematically correct with these people.
&gt;
&gt; I say they are YOU, and they are the modern math world.
&gt;
&gt; You've learned that math is just some way to b.s. people too
&gt; intimidated to check you, while you've accepted things that are
&gt; convenient but false, like the rather naive and wrong ideas that base
&gt; what you call Galois Theory.
&gt;
&gt; Evariste Galois was right, but you people twisted his research.
&gt;
&gt;
&gt; James Harris

Date Subject Author
9/25/04 JAMES HARRIS
9/25/04 David Kastrup
9/26/04 JAMES HARRIS
9/26/04 David Kastrup
9/26/04 David Kastrup
9/27/04 JAMES HARRIS
9/27/04 Andrzej Kolowski
9/27/04 Proginoskes
9/28/04 JAMES HARRIS
9/28/04 Dik T. Winter
9/29/04 magidin@math.berkeley.edu
9/29/04 William Hughes
9/26/04 Tim Smith
9/26/04 Jesse F. Hughes
9/26/04 W. Dale Hall