Can systems of polynomial inequalities be reduced to systems of quadratic inequalities? This Wikipedia article describes a method to solve systems of polynomial inequalities by reducing them to systems of quadratic equations, though I'm not sure if it's accurate:

Firstly, any system of polynomial inequalities can be reduced to a system of quadratic inequalities by increasing the number of variables and equations (for example, by setting a square of a variable equal to a new variable).

This section of the article does not cite any sources, so I can't find any further information about this method. Are there any references that describe this method for solving systems of polynomial inequalities?
 A: If you have an inequality $f(x_1,\ldots,x_n)\ge0$, invent a new variable $y$
and replace by $f(x_1,\ldots,x_n)=y^2$.
If you have an inequality $f(x_1,\ldots,x_n)>0$, invent a new variable $y$
and replace by $f(x_1,\ldots,x_n)y^2=1$.
Therefore you can replace the inequalities by equations.
If you have a monomial like $x_1x_2x_3x_4$ in one of your equations, invent new
variables $y_1$ and $y_2$, add equations $y_1=x_1x_2$, $y_2=y_1x_3$ and replace
$x_1x_2x_3x_4$ in your original equation by $y_2x_4$. Keep doing this to eliminate
monomials of degree $\ge3$.
A: Angina Seng already answered the question in the title of the post. Regarding the other question (in the post and comments), "does this procedure lead to a (more) effective algorithm?", the answer is no. This is because the problem of solving a system of polynomial equations doesn't get any easier when the degrees are bounded by two.
By the way, from glancing at some sources below, the problem seems computationally very hard.
https://en.wikipedia.org/wiki/Gr%C3%B6bner_basis
https://en.wikipedia.org/wiki/Buchberger%27s_algorithm
Solving systems of quadratic equations

One might wish that the degree 2 bound in the inputs could allow a nicer solution, but no: a degree 2 bounded solution algorithm is actually an algorithm for a system with unconstrained degrees.

A general method for solving systems of quadratic equations

The restriction that all polynomial are quadratic is not really a simplification.

https://math.berkeley.edu/~bernd/cbms.pdf
