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

Dr. Moews was able to find a Counter-Example, thus refuting my
Theorems 6 and 7, of my paper arXiv:0803.0018v4

The Counter-Example is the bivariate Polynomial Q = (x1 - x2)^2 + 1.

In this Counter-Example, Q has no real root, and it can be observed
that as x2 tends to infinity, then one has no clear definition of what
is the min. degree of the Univariate Polynomial, which when multiplied
with Q_x2_(x1), would yield a Univariate Polynomial with positive
coefficients.

However, I believe that this problem can be averted for Polynomial Q,
if Q is allowed either of two choices:
CHOICE-1: Q has a real root in a region defined by each variable being
bounded between 0 and some natural number
CHOICE-2: Q does not have a real root in space.

This is important, because the Polynomial derived from 3-SAT, does
fall under the above category. Here, Q is initially defined in
variables X1..Xu, and it is known that the feasible solution of Q
falls in the region defined by each variable being bounded between 0
and 3. So the "something" that needs to be added to Q, is (Yi-Xi+3)^2
+ (Zi+Xi)^2, i running as integers from 1 to u. Here K represents the
max coefficient in Q.

So if we perform the above addition operation on Q, then I believe
that Theorems-6 & 7, will then hold, and also the Conjecture of
F0...FC being continuous ??? Wait, let me see what Dr. Moews says.

Thanks & faithfully,
-Deepak
.