We decompose the copositive cone \(\mathcal {COP}^{n}\) into a disjoint union of a finite number of open subsets \(S_{{\mathcal {E}}}\) of algebraic sets \(Z_{{\mathcal {E}}}\). Each set \(S_{{\mathcal {E}}}\) consists of interiors of faces of \(\mathcal {COP}^{n}\). On each irreducible component of \(Z_{{\mathcal {E}}}\) these faces generically have the same dimension. Each algebraic set \(Z_{{\mathcal {E}}}\) is characterized by a finite collection \({{\mathcal {E}}} = \{(I_{\alpha },J_{\alpha })\}_{\alpha = 1,\dots ,|\mathcal{E}|}\) of pairs of index sets. Namely, \(Z_{{\mathcal {E}}}\) is the set of symmetric matrices A such that the submatrices \(A_{J_{\alpha } \times I_{\alpha }}\) are rank-deficient for all \(\alpha \). For every copositive matrix \(A \in S_{{\mathcal {E}}}\), the index sets \(I_{\alpha }\) are the minimal zero supports of A. If \(u^{\alpha }\) is a corresponding minimal zero, then \(J_{\alpha }\) is the set of indices j such that \((Au^{\alpha })_j = 0\). We call the pair \((I_{\alpha },J_{\alpha })\) the extended support of the zero \(u^{\alpha }\), and \({{\mathcal {E}}}\) the extended minimal zero support set of A. We provide some necessary conditions on \({{\mathcal {E}}}\) for \(S_{{\mathcal {E}}}\) to be non-empty, and for a subset \(S_{{{\mathcal {E}}}'}\) to intersect the boundary of another subset \(S_{{\mathcal {E}}}\).

我们将正圆锥\（\ mathcal {COP} ^ {n} \）分解成有限个开放子集\（S _ {{\ mathcal {E}}}} \的代数集\ {Z_ { {\ mathcal {E}}} \）。每个集合\（S _ {{\ mathcal {E}}} \）由\（\ mathcal {COP} ^ {n} \）的面孔内部组成。在\（Z _ {{\ mathcal {E}}} \）的每个不可约分量上，这些面通常具有相同的尺寸。每个代数集\（Z _ {{\ mathcal {E}}} \）的特征在于有限集合\（{{\ mathcal {E}}} = \ {（I _ {\ alpha}，J _ {\ alpha}） \} _ {\ alpha = 1，\ dots，| \ mathcal {E} |} \）对索引集。即\（Z _ {{\ mathcal {E}}} \）是对称矩阵A的集合，这样子矩阵\（A_ {J _ {\ alpha} \ times I _ {\ alpha}} \）对于所有\（\ alpha \）都是秩不足的。对于每个正定矩阵\（S _ {{{\ mathcal {E}}} \}中的A，索引集\（I _ {\ alpha} \）是A的最小零支持。如果\（u ^ {\ alpha} \）是对应的最小零，则\（J _ {\ alpha} \）是索引j的集合，使得\（（Au ^ {\ alpha}）_ j = 0 \）。我们称这对\（（I _ {\ alpha}，J _ {\ alpha}）\）为零\（u ^ {\ alpha} \）和\（{{\ mathcal {E}}}的扩展支持。\）扩展最小零支撑集的甲。我们在\（{{\ mathcal {E}}} \\）上为\（S _ {{\ mathcal {E}}} \\）非空以及子集\（S _ {{{ \ mathcal {E}}}'} \）与另一个子集\（S _ {{\ mathcal {E}}} \）的边界相交。