# The existential theory of the reals

Some definitions of the existential theory of the reals (ETR) allow a real closed field and some definitions allow only rational numbers as coefficients of polynomials. Which one is correct? Will the answer to the question have an effect on the PSPACE proof by J. Canny? For example, ETR is not in PSPACE, if numbers in real closed field are allowed as coefficients?

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For each coefficient, you introduce one variable, an equation stating the variable is a root of the respective polynomial, and something to uniquely pick up the particular root: for example, if the coefficient is specified as the unique root of $f(x)$ in the interval $[a,b]$, you would include the inequalities $x-a\ge0$ and $b-x\ge0$ (which can be combined to $(x-a)(b-x)\ge0$). This is a polynomial-time reduction, the description of the new system has length polynomial in the length of the original input (in effect, it is the same input organized in a different way). ... –  Emil Jeřábek Mar 8 '13 at 17:06