# Transformation of sets defined by polynomial inequalities in a subspace to another subspace

Given a set defined by polynomial inequalites in one subspace of $$\mathbb{R}^n$$, is it possible to determine polynomial inequalites defining the set in another subspace of $$\mathbb{R}^n$$?

The first subspace is of form $$\mathbf{u}=U^T\mathbf{x}$$ and the second is of form $$\mathbf{v}=V^T\mathbf{x}$$, where $$U \in \mathbb{R}^{n \times d_1}$$ and $$V \in \mathbb{R}^{n \times d_2}$$ have orthonormal columns.

For the first subspace, a set is defined such that some polynomial constraints are satisfied.

$$\Omega_u = \{ \mathbf{u} | h_i(\mathbf{u}) \leq b_i \}$$

I would like to know if if it is possible (and if so, what literature would be useful to read) to define the set

$$\Omega_v = \{ \mathbf{v} | h_i(\mathbf{u}) \leq b_i \}$$

by polynomial inequalities in the subspace $$\mathbf{v}$$

$$\Omega_v = \{ \mathbf{v} | g_i(\mathbf{v}) \leq c_i \}$$

such that these new polynomial inequalites $$g_i(\mathbf{v}) \leq c_i$$ completely define the set $$\Omega_v$$.

In the case where $$h_i(\mathbf{u})$$ and $$g_i(\mathbf{v})$$ are linear, this problem is the same as finding the projection of a polytope (first lift the set $$\Omega_u$$ to the full $$n$$-dimensional space, and then project onto $$d_2$$-dimensional space). However, I am interested in the cases in which these functions are nonlinear, particularly when they are polynomials of order 2 or above.