--- title: Strongly Robust Simplicial Complex url: https://www.emergentmind.com/topics/strongly-robust-simplicial-complex type: topic --- # Strongly Robust Simplicial Complex Searching arXiv for recent and directly relevant papers on “strongly robust simplicial complex” and adjacent formulations. {"query":"all:\"strongly robust simplicial complex\" OR title:\"strongly robust simplicial complex\" OR abs:\"strongly robust simplicial complex\"","max_results":10} {"query":"\"strongly robust simplicial complex\" arXiv","max_results":10} One formal usage of **strongly robust simplicial complex** arises in the toric-ideal literature: for a simple toric ideal \(I_T\), the strongly robust simplicial complex \(\Delta_T\) is the simplicial complex whose faces record exactly which bouquet-signature patterns yield strongly robust toric ideals having bouquet ideal \(I_T\). In that setting, \(\Delta_T\) is a combinatorial classifier for strong robustness, expressed through Graver bases, indispensable binomials, and bouquet decompositions. In the broader simplicial-complex literature, however, “robustness” also appears in distinct senses, notably \(r\)-ampleness, behavior under strong collapse, and \(k\)-robust clique complexes; these are related notions rather than the same invariant [2206.03121]. ## 1. Bouquet data, simple toric ideals, and strong robustness Let \(A=[{\bf a}_1,\dots,{\bf a}_n]\in\mathbb{Z}^{m\times n}\) with \(\ker_{\mathbb{Z}}(A)\cap\mathbb{N}^n=\{{\bf 0}\}\). The toric ideal is \[ I_A=\big\langle x^{{\bf u}^+}-x^{{\bf u}^-}\mid {\bf u}\in\ker_{\mathbb{Z}}(A)\big\rangle \subset K[x_1,\dots,x_n]. \] Its oriented-matroid side is encoded by the bouquet graph \(G_A\), whose connected components are the **bouquets**. Bouquets are **free**, **mixed**, or **non-mixed**. A toric ideal is **simple** when every bouquet is a singleton. From the bouquets one constructs the bouquet ideal \(I_{A_B}\), and \(I_{A_B}\) is always simple. The strong robustness condition is formulated in terms of canonical generating sets. A toric ideal is **strongly robust** when the Graver basis is a minimal generating set. Equivalently, the set of indispensable binomials, any minimal binomial generating set, any reduced Gröbner basis, the universal Gröbner basis, and the Graver basis all coincide. This is the strongest possible coincidence among the standard Gröbner-Markov-Graver objects attached to a toric ideal [2206.03121]. ## 2. Definition of the strongly robust simplicial complex Fix a simple toric ideal \[ I_T\subset K[x_1,\dots,x_s]. \] For \(\omega\subseteq [s]\), a toric ideal \(I_A\) is called a **\(T_\omega\)-ideal** or **\(T_\omega\)-robust** when its bouquet ideal is \(I_T\) and \(\omega\) is exactly the set of indices of non-mixed bouquets. For such a class, one transports indispensability from \(I_A\) back to \(\ker_{\mathbb{Z}}(T)\): the \(T_\omega\)-indispensable set \(S_\omega(T)\) consists of those Graver elements of \(T\) that become indispensable after the bouquet expansion. The strongly robust simplicial complex is then \[ \Delta_T=\{\omega\subseteq [s]\mid S_\omega(T)=\Gr(T)\}. \] Thus \(\omega\in\Delta_T\) precisely when **every** Graver element of \(I_T\) becomes indispensable for bouquet pattern \(\omega\). This immediately makes \(\Delta_T\) the combinatorial record of which non-mixed bouquet patterns are compatible with strong robustness. The defining theorem is: if \(I_A\) is a \(T_\omega\)-robust toric ideal, then \[ I_A\text{ is strongly robust}\quad\Longleftrightarrow\quad \omega\in\Delta_T. \] A second equivalent criterion uses the second Lawrence lifting. For \[ \Lambda(T)=\begin{pmatrix}T&0\\ I_s&I_s\end{pmatrix}, \] let \(\Lambda(T)_\omega\) be obtained by deleting the \((m+i)\)-th row and the \((s+i)\)-th column for each \(i\in\omega\). Then \[ \omega\in\Delta_T \quad\Longleftrightarrow\quad I_{\Lambda(T)_\omega}\text{ is strongly robust}. \] Since \(S_{\omega_1}(T)\supseteq S_{\omega_2}(T)\) whenever \(\omega_1\subseteq \omega_2\), the family \(\Delta_T\) is downward closed, hence is indeed a simplicial complex [2206.03121, 2510.04730]. ## 3. Dimension bounds and structural restrictions The geometry of \(\Delta_T\) is highly constrained. In codimension \(2\), the reduced Gale transform gives a centrally symmetric polygon \(P\), and \(\Delta_T\) is a simplicial subcomplex of the simplex whose vertices are exactly those indices \(i\) for which \(\widetilde{\bf t}_i\) is **not** a vertex of \(P\). In the same codimension, robustness and strong robustness coincide: \[ I_A\text{ robust}\quad\Longleftrightarrow\quad I_A\text{ strongly robust}. \] Accordingly, in codimension \(2\), \(\Delta_T\) controls robustness as well as strong robustness. A different global restriction appears for simple configurations in general position. If \(T\) is a configuration in general position, then \[ \dim(\Delta_T)<\operatorname{rank}(T). \] This gives a partial answer to Sullivant’s question. The bound is sharp: there exist simple toric ideals of rank \(m\) with \[ \dim(\Delta_T)=m-1. \] An explicit cyclic example is \[ T^5_{[7]}= \begin{pmatrix} 1&1&1&1&1&1&1\\ 1&2&3&4&5&6&7\\ 1&4&9&16&25&36&49\\ 1&8&27&64&125&216&343\\ 1&16&81&256&625&1296&2401 \end{pmatrix}, \] for which \[ \Delta_{T^5_{[7]}}=2^{\{1,3,4,5,7\}},\qquad \dim(\Delta_{T^5_{[7]}})=4. \] Since configurations in general position are simple and satisfy \(\dim(\Delta_T)<\operatorname{rank}(T)\), their toric ideals are never strongly robust [2206.03121, 2510.04730]. ## 4. Monomial curves For monomial curves, the structure simplifies drastically. If \[ T=(n_1,\dots,n_s)\in\mathbb{Z}^{1\times s},\qquad s\ge 3, \] then the toric ideal \(I_T\) is simple. The strongly robust simplicial complex of a monomial-curve ideal satisfies a severe restriction: \[ \Delta_T\text{ is either }\{\emptyset\}\text{ or contains exactly one 0-dimensional face.} \] Equivalently, \(\Delta_T\) is either \(\{\emptyset\}\) or \(\{\emptyset,\{i\}\}\) for a unique \(i\in[s]\). In \(\mathbb{A}^3\), the classification is explicit. If \(T=(n_1,n_2,n_3)\), then: - if \(I_T\) is **not** a complete intersection, \(\Delta_T\) is the empty complex; - if \(I_T\) is a complete intersection on \(n_i\), then \[ \Delta_T=\{\emptyset,\{i\}\}; \] - if \(I_T\) is a complete intersection on all, then \(\Delta_T\) is the empty complex. This yields the sharp criterion: \[ \Delta_T\text{ contains one 0-dimensional face} \quad\Longleftrightarrow\quad I_T\text{ is a complete intersection ideal with exactly two Betti degrees.} \] For monomial curves with \(s\ge 3\), the ideal \(I_T\) itself is never strongly robust, because its bouquet pattern has all bouquets non-mixed, while \(\Delta_T\) has dimension at most \(0\) [2305.11743]. ## 5. Generalized Lawrence constructions and classification The monomial-curve case admits a complete constructive description. Given \[ T=(n_1,\dots,n_s) \] and integer vectors \[ c_j=(c_{j1},\dots,c_{jm_j})\in\mathbb{Z}^{m_j}, \] with full support and \(\gcd(c_{j1},\dots,c_{jm_j})=1\), one forms a generalized Lawrence matrix \[ A= \begin{pmatrix} A_1&0&\cdots&0\\ 0&A_2&\cdots&0\\ \vdots&&\ddots&\vdots\\ 0&0&\cdots&A_s\\ C_1&C_2&\cdots&C_s \end{pmatrix}, \] where \(A_j=A(n_j,c_j)\) and \(C_j=C(c_j)\). The sign pattern of the \(c_j\) determines whether the corresponding bouquet is mixed or non-mixed. If \(\Delta_T=\{\emptyset\}\), all \(c_j\) are chosen with first coordinate positive and at least one negative coordinate, forcing all bouquets to be mixed. If \(\Delta_T=\{\emptyset,\{i\}\}\), the same holds for \(j\neq i\), while \(c_i\) may be all positive. With these sign conditions, \(I_A\) is strongly robust and has bouquet ideal \(I_T\). Moreover, this construction is exhaustive: every strongly robust toric ideal whose bouquet ideal is the ideal of a monomial curve is produced by such a generalized Lawrence matrix. Consequently, whenever a monomial-curve bouquet ideal admits strongly robust realizations, it admits infinitely many of them, and they are all classified by this Lawrence-type construction [2305.11743]. ## 6. Other uses of “robustness” in simplicial-complex theory Outside toric geometry, robustness is formalized differently. One major notion is **\(r\)-ampleness**. A simplicial complex \(X\) is \(r\)-ample if for every \(U\subseteq V(X)\) with \(|U|\le r\) and every subcomplex \(A\subseteq X_U\), there exists \(v\in V(X)\setminus U\) such that \[ \mathrm{lk}_X(v)\cap X_U=A. \] The countable \(\infty\)-ample Rado complex is “totally indestructible”: removing any finite number of simplices leaves a complex isomorphic to itself. Finite \(r\)-ample complexes are finite approximations to that behavior, and if \(Y\) is obtained from an \(r\)-ample complex \(X\) by deleting a finite family \(\mathcal F\) of simplices, then \(Y\) remains \((r-k)\)-ample provided \[ |\mathcal F|+\dim(\mathcal F)