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

There is an important concept that is worth mention here, regarding my
Proofs of Theorem-6 and 7.

Consider a u-variate Q. I have basically stated in my proof of Theorem
6 and 7, that Q can be multiplied by F = F0 + F1(X) + F2(X^2) + ... +
FD (X^C), will yield a Polynomial E = E0 + E1(X1) + E2(X1^2) + ... +
ED(X1^D), where each of {F0,F1,F2...FC} and {E0,E1,E2...ED} denote
Polynomials in {X2,X3...Xu}, and where where each of {F0,F1,F2...FC}
and {E0,E1,E2...ED} are greater than 0, for all real positive values
of {X2,X3...Xu}.

A doubt that some of you might get is, whether it is correct to say
that each of {F0,F1,F2...FC} and {E0,E1,E2...ED} denote Polynomials in
{X2,X3...Xu}. Some of you might feel that why can't each of
{F0,F1,F2...FC} and {E0,E1,E2...ED} denote some discontinuous
functions, or why can't they denote continuous functions with infinite
degree example sin(x) or e(-x) ???

In this post, I would like to clear up these potential doubts. I shall
first prove that each of {F0,F1,F2...FC} and {E0,E1,E2...ED} can be
identified to be continuous functions. Interestingly and separately,
they can also be identified to be discontinous functions. I shall then
go on to prove that when they identified as continuous functiont, then
they do not need components of functions with infinite degree. So here
goes:

Let Q be a u-variate Polynomial in {X1,X2,...Xu}, that has no real
root in space. Let L denote the family of univariate Polynomials
obtainable in X1, by putting the other variables in Q as any Real
number. Let U denote a univariate Polynomial, whose degree is FIXED at
the worst case minimum degree required for a univariate Polynomial,
which when multiplied with any member of family L, will produce a
univariate Polynomial with positive coefficients. Now, if we
continuously vary the other variables, i.e. {X2,X3,...Xu} along the
Real number line, then it is possible to observe that the real
coefficients of U will also vary continuously, such that U multiplied
by the corresponding member of L, will yield a univariate Polynomial
with positive coefficients. Due to this continuous behavior, it is
correct to state that Q can be multiplied by F = F0 + F1(X1) +
F2(X1^2) + ... + FD (X1^C), will yield a Polynomial E = E0 + E1(X1) +
E2(X1^2) + ... + ED(X1^D), where each of {F0,F1,F2...FC} and
{E0,E1,E2...ED} denote continuous functions in {X2,X3...Xu}, and where
where each of {F0,F1,F2...FC} and {E0,E1,E2...ED} are greater than 0,
for all real positive values of {X2,X3...Xu}. Here C is fixed at the
constant WORST CASE MIN DEGREE, that I defined in this same paragraph.

It is important to note that in my above paragraph, it possible to
identify both continuous function definitions AS WELL AS discontinuous
function definitions of {F0,F1,F2...FC} and {E0,E1,E2...ED}. A
discontinuous function definition would be obtained, if one follows
the practice of using U only the least degree required, which when
multiplied with a member of family L, would yield a univariate
polynomial with positive coefficients (so in this case the degree of U
is dynamic and depends on the member of family L). However, if we hold
the degree of U FIXED at the WORST CASE MIN DEGREE, then one can get
continuous function definitions of {F0,F1,F2...FC} and
{E0,E1,E2...ED}.

Now that we proved that it is possible to identify {F0,F1,F2...FC} and
{E0,E1,E2...ED} as continuous functions, I shall now go on to prove
that only Polynomial functions (and not functions with inf degree like
sin(x) or e(-x)) are sufficient to be a part of our definitions of
{F0,F1,...FC} and {E0,E1,E2...ED}. Here goes:

We can write Q = Q0 + Q1(X1) + Q2(X1^2) + ... + QB(X1^B), where
{Q0,Q1,...QB} represent Polynomials in {X2,X3...Xu}. And we also
earlier wrote F = F0 + F1(X) + F2(X^2) + ... + FD (X^C). Now consider
the product of Q and F. Since each coefficient of this product needs
to be greater than zero, we have the following (B+C+1) equations,
assuming Q0=1:

F0 > 0
F0 Q1 + F1 > 0
F0 Q2 + F1 Q1 + F2 > 0
F0 Q3 + F1 Q2 + F2 Q1 + F3 > 0
....
....

These can be re-written as follows:
F0 = EPSILON_1
F1 = EPSILON_2 - F0 Q1
F2 = EPSILON_3 - F0 Q2 - F1 Q1
....
....
where EPSILON_1, EPSILON_2, EPSILON_3, etc, represent positive
quantities (that are not necessarily small).

Now, as each of {Q0,Q1...QB} is a Polynomial function (and not any
other function of Infinite degree), therefore, it follows that it
should be possible to identify {F0,F1,...FC} and {E0,E1,E2...ED} that
are also Polynomial functions.

Thanks & faithfully,
-Deepak
.