The Arveson boundary of a free quadrilateral is given by a noncommutative variety

Let $SM_n(\mathbb{R})^g$ denote $g$-tuples of $n \times n$ real symmetric matrices and set $SM(\mathbb{R})^g = \cup_n SM_n(\mathbb{R})^g$. A free quadrilateral is the collection tuples $X \in SM(\mathbb{R})^2$ which have positive semidefinite evaluation on linear equations defining a classical quadrilateral. Such closed under rich class convex combinations called matrix combination. That is, given elements $X=(X_1, \dots, X_g) SM_{n_1}(\mathbb{R})^g$ $Y=(Y_1, Y_g) SM_{n_2}(\mathbb{R})^g$ $\mathcal{Q}$, one has \[ V_1^T X V_1+V_2^T Y V_2 \mathcal{Q} \] for any contractions $V_1:\mathbb{R}^n \to \mathbb{R}^{n_1}$ $V_2:\mathbb{R}^n \mathbb{R}^{n_2}$ satisfying $V_1^T V_2=I_n$. These are natural analogue in dimension setting. extreme point quadrilaterals points: cannot be expressed as nontrivial combination points serve minimal recovers through combinations. In this article we show that determined by zero noncommutative polynomials. More precisely, construct polynomials $p_1,p_2,p_3,p_4$ such tuple SM (\mathbb{R})^2$ $\mathcal{Q}$ if only \mathcal{Q}$ $p_i(X) =0 $ $i=1,2,3,4$ $X$ irreducible. addition establish several basic results projective maps spectrahedra homogeneous spectrahedra.