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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.

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