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

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, 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 + (0.01)^2, 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, 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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