What I will ask, more than a solution, is better mathematical definition of my problem and directions to find the solution.

I have a set of linear equations, e.g.:

\begin{align} d_1 &= L_1 - 9\,m_1 - 9\,m_2 \\ d_2 &= x_1 + 3\,m_1 + 3\,m_2 \\ d_3 &= y_1 \\ d_4 &= L_2 - 4\,m_2 \\ d_5 &= x_2 + 2\,m_2 \\ d_6 &= y_2 \end{align}

where $d_1,d_2,...,d_6$ are linear combinations of $L_1,x_1,y_1,m_1,L_2,x_2,y_2,m_2 \in \mathbb{R}$.

$L_1,x_1,y_1,m_1,L_2,x_2,y_2,m_2$ are physical parameters which have physical nonlinear constraints, in the form of polynomial inequalities, e.g.:

\begin{align} m_1 &>0\\ L_1m_1−x^2_1−y^2_1 &>0\\ m_2 &>0\\ L_2m_2−x^2_2−y^2_2 &>0 \end{align}

I would like to rewrite constraints in terms of $d_1,d_2,...,d_6$ only.

I.e., I would like to find constrains over $d_1,d_2,...,d_6$ parameters (only), so that when a numerical set of $d_1,d_2,...,d_6$ verifies the new mapped constraints that would mean that there is at least one $L_1,x_1,y_1,m_1,L_2,x_2,y_2,m_2$ solution (it doesn't matter what) which verify the former constraints. If the new constraints are not verified it must mean that no $L_1,x_1,y_1,m_1,L_2,x_2,y_2,m_2$ solution exists.

Here I presented a particularly small example, for it I was already able to find the constraints doing manual equation manipulation ($d_1 + 6 d_2 > 0$ and $- 9 d_{5}^{2} - 9 d_{6}^{2} + \left(d_{1} + 6 d_{2}\right) \left(d_{4} + 4 d_{5}\right) >0$) However, I have problems with up to 70 linear equations and 30 higher order polynomial inequalities constraints.

What I need is a systematic method to write the constrains over $d_1,d_2,...,d_6$.

I think that in geometrical thinking this is like projecting the intersection of a set of non-linear volumes/regions of $n$-dimensional space ($n>>1$) onto a subspace of it, where the coefficients of the linear system are the basis of such subspace.

Now the questions:

- What kind of problem do I have?
- Which mathematical fields shall I study, and which directions must I follow?

My background is in engineering so my mathematical writing is not very formal.

Thanks.