Can systems of polynomial inequalities be reduced to systems of quadratic inequalities?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).

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g2joey15 · Accepted Answer

If you have an inequality   f  (      x    1    ,  …  ,      x    n    )  ≥  0, invent a new variable   y and replace by   f  (      x    1    ,  …  ,      x    n    )  =      y    2  .If you have an inequality   f  (      x    1    ,  …  ,      x    n    )  &amp;gt;  0, invent a new variable   y and replace by   f  (      x    1    ,  …  ,      x    n    )      y    2    =  1Therefore you can replace the inequalities by equations.If you have a monomial like       x    1        x    2        x    3        x    4   in one of your equations, invent new variables       y    1   and       y    2  , add equations       y    1    =      x    1        x    2  ,       y    2    =      y    1        x    3   and replace       x    1        x    2        x    3        x    4   in your original equation by       y    2        x    4  . Keep doing this to eliminate monomials of degree   ≥  3.

Can systems of polynomial inequalities be reduced to systems of

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