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

MY PREVIOUS POST WAS MADE IN ERROR, SO PLS DISCARD MY PREVIOUS POST,
AND READ THE BELOW CORRECTED POST:

To give an example, for the bivariate case, consider Q(x1,x2) =
(x1-1)^2 + (0.01)^2 + (x2)^2.

Now when Q(x1,0) = (x1-1)^2 + (0.01)^2, we will need a univariate
Polynomial U of minimum degree = INT(1+(100)PI), which when multiplied
with Q(x1,0), will yield a univariate Polynomial with positive real
coefficients. Now, when Q(x1,sqrt(0.9999)) = (x1-1)^2 + 1, then there
is nothing wrong in thinking that one would need a univariate U of
degree = INT(1+PI), in which case there is a discontinuous behavior in
the values of the real coefficients of U, as the coefficient (of say
x1^10) will be seen to suddenly jump from 0 to a large number, when we
increase/decrease x2 gradually. However, what I am saying is that let
us continue to use U of the same degree as used earlier, i.e. degree
of INT(1+PI(100)). Now keeping this degree of U constant at
INT(1+PI(100)), if we were to vary 'x2' continuously starting from 0
and going to say 1000, then you must agree that it is possible for the
coefficients of U to also vary continuously, so that product of U with
the Q_x2_(x1) will yield a univariate Polynomial with positive
coefficients.
.

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