Re: Another approach to decide on existence of a real root for Univariate Polynomials with Integer Coefficients, and a possible Multivariate extension for 3-SAT

I just want to inform you, that I will certainly be able to finish my
Version_5 of my arXiv:0803.0018 paper, before end of August 2008. I
promise that I will not delay any more.

You may just note that in my Version_5, my Theorems 6 & 7 will be
removed (as Dr Moews has refuted them), and they will be replaced by a
single Theorem tailored for 3-SAT as follows:

***** THEOREM BEGINS *****
P = Q P1 + P2 P3, is a Polynomial with positive real coefficients, if
and only if, Q does not have a real root.
***** THEOREM ENDS *****

In my above Theorem, P1, P2, and P3 are Polynomials, and P2 =
SUMMATION ((Xi – 1)^2 (Xi – 2)^2, over all i as integers in [1,u]),
and also Q is the Polynomial derived from the 3-SAT instance as per
the following 3-step procedure.

Step-1: Take one real variable for each Boolean variable in the given
3-SAT instance.
Step-2: Express each clause by a Polynomial. If a clause involves
variables X1, X2, and X3, express it by the polynomial (X1 - i)^2 (X2
- j)^2 (X3 - k)^2, where i is 1 if X1 appears negated and 2 otherwise,
j is 1 if X2 appears negated and 2 otherwise, and k is 1 if X3 appears
negated and 2 otherwise.
Step-3: Set Q = SUMMATION (the Polynomial expressing each clause in
Step 2, over all clauses in the 3-SAT instance).

Linear Programming techniques are an option, to use, to determine the
existance of the Polynomials P1 and P3.

I am not attaching the explanation for my Theorem in this post,
because it will become very messy (mainly because my client does not
allow rich text so I cannot depict subscripts & superscript properly),
but I promise that it will be explained very elaborately in my
Version_5 paper.

Still, if you would like to hear the reason for my proof before end of
August-2008, then please email me, and I will send you a separate
Microsoft Word document, explaining it.

Thanks & faithfully,
-Deepak
.

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