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

The purpose of my argument with the Equations was not meant to be
considered alone, because if considered alone, then they will not
sound convincing.

Let me explain. My argument consists of 2 parts:
PART-1: Here we argue that {F0...FC} can be identified by some
continuous functions, if C is fixed to the constant
WORST_CASE_MIN_DEGREE that I defined in paragraph 5 of my second post
in my public Thread.
PART-2: Here we argue that once they have been identified as
continuous functions, then {F0...FC} do not need components of
functions with infinite degree. By infinite degree, I mean functions
like sqrt(1+(x2)^2), sin(x2), e^(-x2), etc.

I will use an example, i.e. by taking b = c = 3:
F0 Q0 > 0,
F1 Q0 + F0 Q1 > 0,
F2 Q0 + F1 Q1 + F0 Q2 > 0,
F3 Q0 + F2 Q1 + F1 Q2 + F0 Q3 > 0,
F3 Q1 + F2 Q2 + F1 Q3 > 0,
F3 Q2 + F2 Q3 > 0,
F3 Q3 > 0.

which can be re-written as:
F0 Q0 = eps1,
F1 Q0 + F0 Q1 = eps2,
F2 Q0 + F1 Q1 + F0 Q2 = eps3,
F3 Q0 + F2 Q1 + F1 Q2 + F0 Q3 = eps4,
F3 Q1 + F2 Q2 + F1 Q3 = eps5,
F3 Q2 + F2 Q3 = eps6,
F3 Q3 = eps7.

which can again be re-written as:
F0 = (eps1)/Q0.......................................Eq_1
F1 = (eps2 - F0 Q1)/Q0...............................Eq_2
F2 = (eps3 - F1 Q1 - F0 Q2)/Q0.......................Eq_3
F3 = (eps4 - F2 Q1 - F1 Q2 - F0 Q3)/Q0...............Eq_4
F3 = (eps5 - F2 Q2 - F1 Q3)/Q1.......................Eq_5
F3 = (eps6 - F2 Q3)/Q2...............................Eq_6
F3 = (eps7)/Q3.......................................Eq_7


Now, from my PART-I argument, it is obvious that we can find a
definition for {F0...F3} as a set of some continuous functions (these
may or may not be Polynomial functions), where {eps1...eps7} are some
functions that are greater than 0 for all real positive values of
{x1,x2...xu}. Since {F0...F3} are continuous functions, it follows
that {eps1...eps7} are also continuous functions.

Now comes another important part. Eq_1 can be re-written as Eq'_1, by
replacing F1 with F'1 and by replacing eps1 by eps'1, where F'1 is a
purely Polynomial function, and where eps'1 is a continuous function
that is still greater than 0 for all real positive values of
{x1,x2...xu}. The reason why this is possible, is because we can
always transfer the non-Polynomial components from the left-hand-side
to the right-hand-side, and then add some very large arbitrary
Polynomial function components that are always much much greater than
0, on both sides of the Eq_1. These Polynomial functions that are
added on both sides, are chosen so large that they suffocate the
negative effects of the non-Polynomial components that were moved to
the right-hand-side, so that eps'1 is still greater than 0 for all
real positive values of {x1,x2...xu}. Similarly re-write the other
Equations as well. We finally have:

F'0 = (eps'1)/Q0.......................................Eq'_1
F'1 = (eps'2 - F'0 Q1)/Q0...............................Eq'_2
F'2 = (eps'3 - F'1 Q1 - F'0 Q2)/Q0.......................Eq'_3
F'3 = (eps'4 - F'2 Q1 - F'1 Q2 - F'0 Q3)/Q0...............Eq'_4
F'3 = (eps'5 - F'2 Q2 - F'1 Q3)/Q1.......................Eq'_5
F'3 = (eps'6 - F'2 Q3)/Q2...............................Eq'_6
F'3 = (eps'7)/Q3.......................................Eq'_7

Thus, we are able to obtain definitions for {F'0...F'3} as purely
Polynomial functions, and definitions for {eps'1...eps'7} as some
continuous functions, where each of {eps'1...eps'7} are greater than
zero for positive real values of {x1...xu}.

I eagerly await your reply, regarding my above argument.

faithfully,
-Deepak
.