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

Once you agree that the basic idea of my 3 arguments mentioned in my
previous post, are correct, then it should be straightforward to
understand that F0...FC can be identified as Polynomial functions.
Please read my proof for this below:

Our aim is to prove that it is correct to say that a u-variate
Polynomial without a real root, Q = Q0 + Q1 X1 + Q2 X1^2 + ... + QB
X1^B can be multiplied by a u-variate Polynomial F = F0 + F1 X1 + F2
X1^2 + ... + FC X1^C, to yield a u-variate Polynomial E = = E0 + E1 X1
+ E2 X1^2 + ... + ED X1^D, where the following four statements are
true:
1.) B, C and D are some natural numbers, and D=B+C
2.) Fi > 0 for all i as integers in [0,C], and, Ej > 0 for all j as
integers in [0,D], for all {X2,X3...Xu} as positive Reals
3.) Fi is a Polynomial in variables {X2,X3...Xu} for all i as integers
in [0,C], and Ej is a Polynomial in variables {X2,X3...Xu} for all j
as integers in [0,D]
4.) C = WORST_CASE_MIN + LAMBDA. Here LAMBDA is a natural number
greater than 1. Also WORST_CASE_MIN is the worst case minimum degree
required of a univariate Polynomial in X1, with positive coefficients,
which when multiplied with any member of family L, will yield another
univariate Polynomial with positive coefficients. The family L is
defined as the family of all univariate Polynomials in X1 obtainable
from Q, by substituting {X2,X3...Xu} as any real positive number.

Our proof proceeds as follows. Let U denote a univariate Polynomial in
X1 having positive coefficients, whose degree is fixed at C. Now, if
we continuously vary the other variables, i.e. {X2,X3,...Xu} along the
Real number line, keeping the degree of U fixed at C, then it is
possible to observe that the real coefficients of U can 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 mentioned in the above paragraph, it
is correct to state that Q can be multiplied by a u-variate function G
= G0 + G1 X1 + G2 X1^2 + ... + GC X1^C, so as to yield another u-
variate function H = H0 + H1 X1 + H2 X1^2 + ... + HD X1^D, where the
following two statements are true:
1.) Gi is a continuous function in variables {X2,X3,...Xu} for all i
as integers in [1,C], and Hj is a continuous function in variables
{X2,X3,...Xu} for all j as integers in [1,D]
2.) Gi > 0 for all i as integers in [0,C], and, Hj > 0 for all j as
integers in [0,D], for all {X2,X3...Xu} as positive Reals

Now in the product Q G = H, let us plug in the values of Q, G, and H,
as follows:
Q = Q0 + Q1 X1 + Q2 X1^2 + ... + QB X1^B
G = G0 + G1 X1 + G2 X1^2 + ... + GC X1^C
H = H0 + H1 X1 + H2 X1^2 + ... + HD X1^D

Now, since the product Q G = H, is already being satisfied by virtue
of each of {G0,G1...GC} and {H0,H1,...HD} being continuous functions,
and each of {Q0,Q1...QB} being Polynomial functions, therefore it
follows that in the product Q G = H, we can say that the following 2
statements are true:
1.) The Polynomial components on the LHS (Left hand side) of Q G = H,
must equal the Polynomial components on the RHS
2.) The non-Polynomial components on the LHS, must equal the non-
Polynomial components on the RHS

Now we can always add some Polynomial components (that are always
positive) on both the LHS and RHS of Q G = H, which suffocate any non-
Polynomial component on either LHS or RHS that is negative at any
point in space. Note that these newly added Polynomial components are
"absorbable" into any of {G0,G1...GC} or into any of {H0,H1...HD}. Now
remove all non-Polynomial components from both LHS and RHS, and let us
denote the corresponding coefficients of X1, as {F0,F1...FC} instead
of {G0,G1...GC}, and {E0,E1...ED} instead of {H0,H1...HD}.

It is obvious, that the LHS and RHS now contain purely Polynomial
components. Hence Proved.

Thanks & faithfully,
-Deepak
.

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