Check if a symbolic polynomial is positive

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How did you define x and y? If you simply use

sage: x, y = var('x, y')

then Sage has now way of knowing what is a variable and what is a coefficient in you polynomial. Still, for one variable it is possible to do:

sage: p = 4*x^2+2*x+1
sage: all([coef>=0 for coef, degree in p.coefficients(x)])
True

For multivariate polynomials you should use PolynomialRing:

sage: PXY.<x, y> = PolynomialRing(QQ)
sage: p = x^2+y+2*x*y
sage: all([coef>=0 for coef in p.coefficients()])
True