Strongly Robust Simplicial Complex

• The strongly robust simplicial complex is defined as the set of bouquet patterns for which every Graver element becomes indispensable, unifying key generators of toric ideals.
• It acts as a combinatorial classifier that links bouquet decompositions with strong robustness, providing clear criteria for when toric ideals are strongly robust.
• The invariant offers actionable insights into structural limitations, dimension bounds, and applications in contexts such as monomial curves and generalized Lawrence constructions.

Searching arXiv for recent and directly relevant papers on “strongly robust simplicial complex” and adjacent formulations. {"^^^^^^^^4query4 robust simplicial complex\"^^^^^^^^4 OR title:\4^^^^^^^^"strongly robust simplicial complex\"^^^^^^^^4 OR abs:\4^^^^^^^^"strongly robust simplicial complex\"","max_results":^^^^4all:\4query4^^^^} {"^^^^^^^^4query4 robust simplicial complex\" arXiv","max_results":^^^^4all:\4query4^^^^} One formal usage of strongly robust simplicial complex arises in the toric-ideal literature: for a simple toric ideal PRESERVED_PLACEHOLDER_^^^^4query4^^^^, the strongly robust simplicial complex PRESERVED_PLACEHOLDER_^^^^4all:\4^^^^ is the simplicial complex whose faces record exactly which bouquet-signature patterns yield strongly robust toric ideals having bouquet ideal PRESERVED_PLACEHOLDER_^^^^4 OR title:\4^^^^. In that setting, PRESERVED_PLACEHOLDER_^^^^4 OR abs:\4^^^^ 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 (&&&^^^^4query4^^^^&&&).

^^^^4all:\4^^^^. 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 PRESERVED_PLACEHOLDER_^^^^4all:\4query4^^^^, and PRESERVED_PLACEHOLDER_^^^^4all:\4all:\4^^^^ 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 (&&&^^^^4query4^^^^&&&).

^^^^4 OR title:\4^^^^. Definition of the strongly robust simplicial complex

Fix a simple toric ideal

PRESERVED_PLACEHOLDER_^^^^4all:\4 OR title:\4^^^^

For PRESERVED_PLACEHOLDER_^^^^4all:\4 OR abs:\4^^^^, a toric ideal PRESERVED_PLACEHOLDER_^^^^4all:\44^^^^ is called a PRESERVED_PLACEHOLDER_^^^^4all:\4^^^^5-ideal or PRESERVED_PLACEHOLDER_^^^^4all:\4^^^^6-robust when its bouquet ideal is PRESERVED_PLACEHOLDER_^^^^4all:\4^^^^7 and PRESERVED_PLACEHOLDER_^^^^4all:\4^^^^8 is exactly the set of indices of non-mixed bouquets. For such a class, one transports indispensability from PRESERVED_PLACEHOLDER_^^^^4all:\4^^^^9 back to PRESERVED_PLACEHOLDER_^^^^4 OR title:\4query4^^^^: the PRESERVED_PLACEHOLDER_^^^^4 OR title:\4all:\4^^^^-indispensable set PRESERVED_PLACEHOLDER_^^^^4 OR title:\4 OR title:\4^^^^ consists of those Graver elements of PRESERVED_PLACEHOLDER_^^^^4 OR title:\4 OR abs:\4^^^^ that become indispensable after the bouquet expansion.

The strongly robust simplicial complex is then

PRESERVED_PLACEHOLDER_^^^^4 OR title:\44^^^^

Thus PRESERVED_PLACEHOLDER_^^^^4 OR title:\4^^^^5 precisely when every Graver element of PRESERVED_PLACEHOLDER_^^^^4 OR title:\4^^^^6 becomes indispensable for bouquet pattern PRESERVED_PLACEHOLDER_^^^^4 OR title:\4^^^^7. This immediately makes PRESERVED_PLACEHOLDER_^^^^4 OR title:\4^^^^8 the combinatorial record of which non-mixed bouquet patterns are compatible with strong robustness.

The defining theorem is: if PRESERVED_PLACEHOLDER_^^^^4 OR title:\4^^^^9 is a PRESERVED_PLACEHOLDER_^^^^4 OR abs:\4query4^^^^-robust toric ideal, then

PRESERVED_PLACEHOLDER_^^^^4 OR abs:\4all:\4^^^^

A second equivalent criterion uses the second Lawrence lifting. For

PRESERVED_PLACEHOLDER_^^^^4 OR abs:\4 OR title:\4^^^^

let PRESERVED_PLACEHOLDER_^^^^4 OR abs:\4 OR abs:\4^^^^ be obtained by deleting the PRESERVED_PLACEHOLDER_^^^^4 OR abs:\44^^^^-th row and the PRESERVED_PLACEHOLDER_^^^^4 OR abs:\4^^^^5-th column for each PRESERVED_PLACEHOLDER_^^^^4 OR abs:\4^^^^6. Then

PRESERVED_PLACEHOLDER_^^^^4 OR abs:\4^^^^7

Since PRESERVED_PLACEHOLDER_^^^^4 OR abs:\4^^^^8 whenever PRESERVED_PLACEHOLDER_^^^^4 OR abs:\4^^^^9, the family $r$^^^^4query4^^^^ is downward closed, hence is indeed a simplicial complex (&&&^^^^4query4^^^^&&&, &&&^^^^4 OR abs:\4^^^^&&&).

^^^^4 OR abs:\4^^^^. Dimension bounds and structural restrictions

The geometry of $r$^^^^4all:\4^^^^ is highly constrained. In codimension $r$^^^^4 OR title:\4^^^^, the reduced Gale transform gives a centrally symmetric polygon $r$^^^^4 OR abs:\4^^^^, and $r$4 is a simplicial subcomplex of the simplex whose vertices are exactly those indices $r$5 for which $r$6 is not a vertex of $r$7. In the same codimension, robustness and strong robustness coincide: $r$8 Accordingly, in codimension $r$9, $k$^^^^4query4^^^^ controls robustness as well as strong robustness.

A different global restriction appears for simple configurations in general position. If $k$^^^^4all:\4^^^^ is a configuration in general position, then

$k$^^^^4 OR title:\4^^^^

This gives a partial answer to Sullivant’s question. The bound is sharp: there exist simple toric ideals of rank $k$^^^^4 OR abs:\4^^^^ with

$k$4

An explicit cyclic example is

$k$5

for which

$k$6

Since configurations in general position are simple and satisfy $k$7, their toric ideals are never strongly robust (&&&^^^^4query4^^^^&&&, &&&^^^^4 OR abs:\4^^^^&&&).

4. Monomial curves

For monomial curves, the structure simplifies drastically. If

$k$8

then the toric ideal $k$9 is simple. The strongly robust simplicial complex of a monomial-curve ideal satisfies a severe restriction: $$A=[{\bf a}_1,\dots,{\bf a}_n]\in\mathbb{Z}^{m\times n}$$^^^^4query4^^^^ Equivalently, $$A=[{\bf a}_1,\dots,{\bf a}_n]\in\mathbb{Z}^{m\times n}$$^^^^4all:\4^^^^ is either $$A=[{\bf a}_1,\dots,{\bf a}_n]\in\mathbb{Z}^{m\times n}$$^^^^4 OR title:\4^^^^ or $$A=[{\bf a}_1,\dots,{\bf a}_n]\in\mathbb{Z}^{m\times n}$$^^^^4 OR abs:\4^^^^ for a unique $$A=[{\bf a}_1,\dots,{\bf a}_n]\in\mathbb{Z}^{m\times n}$$4.

In $$A=[{\bf a}_1,\dots,{\bf a}_n]\in\mathbb{Z}^{m\times n}$$5, the classification is explicit. If $$A=[{\bf a}_1,\dots,{\bf a}_n]\in\mathbb{Z}^{m\times n}$$6, then:

• if $$A=[{\bf a}_1,\dots,{\bf a}_n]\in\mathbb{Z}^{m\times n}$$7 is not a complete intersection, $$A=[{\bf a}_1,\dots,{\bf a}_n]\in\mathbb{Z}^{m\times n}$$8 is the empty complex;
• if $$A=[{\bf a}_1,\dots,{\bf a}_n]\in\mathbb{Z}^{m\times n}$$9 is a complete intersection on $\ker_{\mathbb{Z}}(A)\cap\mathbb{N}^n=\{{\bf 0}\}$^^^^4query4^^^^, then

$\ker_{\mathbb{Z}}(A)\cap\mathbb{N}^n=\{{\bf 0}\}$^^^^4all:\4^^^^

• if $\ker_{\mathbb{Z}}(A)\cap\mathbb{N}^n=\{{\bf 0}\}$^^^^4 OR title:\4^^^^ is a complete intersection on all, then $\ker_{\mathbb{Z}}(A)\cap\mathbb{N}^n=\{{\bf 0}\}$^^^^4 OR abs:\4^^^^ is the empty complex.

This yields the sharp criterion: $\ker_{\mathbb{Z}}(A)\cap\mathbb{N}^n=\{{\bf 0}\}$4 For monomial curves with $\ker_{\mathbb{Z}}(A)\cap\mathbb{N}^n=\{{\bf 0}\}$5, the ideal $\ker_{\mathbb{Z}}(A)\cap\mathbb{N}^n=\{{\bf 0}\}$6 itself is never strongly robust, because its bouquet pattern has all bouquets non-mixed, while $\ker_{\mathbb{Z}}(A)\cap\mathbb{N}^n=\{{\bf 0}\}$7 has dimension at most $\ker_{\mathbb{Z}}(A)\cap\mathbb{N}^n=\{{\bf 0}\}$8 (Kosta et al., 2023).

5. Generalized Lawrence constructions and classification

The monomial-curve case admits a complete constructive description. Given

$\ker_{\mathbb{Z}}(A)\cap\mathbb{N}^n=\{{\bf 0}\}$9

and integer vectors

$$I_A=\big\langle x^{{\bf u}^+}-x^{{\bf u}^-}\mid {\bf u}\in\ker_{\mathbb{Z}}(A)\big\rangle \subset K[x_1,\dots,x_n].$$^^^^4query4^^^^

with full support and $$I_A=\big\langle x^{{\bf u}^+}-x^{{\bf u}^-}\mid {\bf u}\in\ker_{\mathbb{Z}}(A)\big\rangle \subset K[x_1,\dots,x_n].$$^^^^4all:\4^^^^, one forms a generalized Lawrence matrix

$$I_A=\big\langle x^{{\bf u}^+}-x^{{\bf u}^-}\mid {\bf u}\in\ker_{\mathbb{Z}}(A)\big\rangle \subset K[x_1,\dots,x_n].$$^^^^4 OR title:\4^^^^

where $$I_A=\big\langle x^{{\bf u}^+}-x^{{\bf u}^-}\mid {\bf u}\in\ker_{\mathbb{Z}}(A)\big\rangle \subset K[x_1,\dots,x_n].$$^^^^4 OR abs:\4^^^^ and $$I_A=\big\langle x^{{\bf u}^+}-x^{{\bf u}^-}\mid {\bf u}\in\ker_{\mathbb{Z}}(A)\big\rangle \subset K[x_1,\dots,x_n].$$4. The sign pattern of the $$I_A=\big\langle x^{{\bf u}^+}-x^{{\bf u}^-}\mid {\bf u}\in\ker_{\mathbb{Z}}(A)\big\rangle \subset K[x_1,\dots,x_n].$$5 determines whether the corresponding bouquet is mixed or non-mixed. If $$I_A=\big\langle x^{{\bf u}^+}-x^{{\bf u}^-}\mid {\bf u}\in\ker_{\mathbb{Z}}(A)\big\rangle \subset K[x_1,\dots,x_n].$$6, all $$I_A=\big\langle x^{{\bf u}^+}-x^{{\bf u}^-}\mid {\bf u}\in\ker_{\mathbb{Z}}(A)\big\rangle \subset K[x_1,\dots,x_n].$$7 are chosen with first coordinate positive and at least one negative coordinate, forcing all bouquets to be mixed. If $$I_A=\big\langle x^{{\bf u}^+}-x^{{\bf u}^-}\mid {\bf u}\in\ker_{\mathbb{Z}}(A)\big\rangle \subset K[x_1,\dots,x_n].$$8, the same holds for $$I_A=\big\langle x^{{\bf u}^+}-x^{{\bf u}^-}\mid {\bf u}\in\ker_{\mathbb{Z}}(A)\big\rangle \subset K[x_1,\dots,x_n].$$9, while $G_A$^^^^4query4^^^^ may be all positive.

With these sign conditions, $G_A$^^^^4all:\4^^^^ is strongly robust and has bouquet ideal $G_A$^^^^4 OR title:\4^^^^. 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 (Kosta et al., 2023).

6. Other uses of “robustness” in simplicial-complex theory

Outside toric geometry, robustness is formalized differently. One major notion is $G_A$^^^^4 OR abs:\4^^^^-ampleness. A simplicial complex $G_A$4 is $G_A$5-ample if for every $G_A$6 with $G_A$7 and every subcomplex $G_A$8, there exists $G_A$9 such that

PRESERVED_PLACEHOLDER_^^^^4all:\4query4query4^^^^

The countable PRESERVED_PLACEHOLDER_^^^^4all:\4query4all:\4^^^^-ample Rado complex is “totally indestructible”: removing any finite number of simplices leaves a complex isomorphic to itself. Finite PRESERVED_PLACEHOLDER_^^^^4all:\4query4 OR title:\4^^^^-ample complexes are finite approximations to that behavior, and if PRESERVED_PLACEHOLDER_^^^^4all:\4query4 OR abs:\4^^^^ is obtained from an PRESERVED_PLACEHOLDER_^^^^4all:\4query44^^^^-ample complex PRESERVED_PLACEHOLDER_^^^^4all:\4query4^^^^5 by deleting a finite family PRESERVED_PLACEHOLDER_^^^^4all:\4query4^^^^6 of simplices, then PRESERVED_PLACEHOLDER_^^^^4all:\4query4^^^^7 remains PRESERVED_PLACEHOLDER_^^^^4all:\4query4^^^^8-ample provided

PRESERVED_PLACEHOLDER_^^^^4all:\4query4^^^^9

Random medial-regime simplicial complexes are PRESERVED_PLACEHOLDER_^^^^4all:\4all:\4query4^^^^-ample asymptotically almost surely, and their topological complexity satisfies PRESERVED_PLACEHOLDER_^^^^4all:\4all:\4all:\4^^^^ asymptotically almost surely (Even-Zohar et al., 2020, Farber, 2023).

A second notion concerns strong collapse and strong homotopy. In this setting, PRESERVED_PLACEHOLDER_^^^^4all:\4all:\4 OR title:\4^^^^ is a strong homotopy invariant, but the geometric simplicial LS category is not: there exist complexes PRESERVED_PLACEHOLDER_^^^^4all:\4all:\4 OR abs:\4^^^^ with

PRESERVED_PLACEHOLDER_^^^^4all:\4all:\44^^^^

Thus robustness under strong collapse fails for PRESERVED_PLACEHOLDER_^^^^4all:\4all:\4^^^^5, even though it holds for PRESERVED_PLACEHOLDER_^^^^4all:\4all:\4^^^^6 (&&&^^^^4all:\4query4^^^^&&&).

A third notion is the PRESERVED_PLACEHOLDER_^^^^4all:\4all:\4^^^^7-robust clique complex

PRESERVED_PLACEHOLDER_^^^^4all:\4all:\4^^^^8

whose simplices are vertex sets containing no independent set of size PRESERVED_PLACEHOLDER_^^^^4all:\4all:\4^^^^9. For square sequence graphs and rectangular grids, PRESERVED_PLACEHOLDER_^^^^4all:\4 OR title:\4query4^^^^ has homotopy type a wedge of spheres; in particular, for PRESERVED_PLACEHOLDER_^^^^4all:\4 OR title:\4all:\4^^^^ and PRESERVED_PLACEHOLDER_^^^^4all:\4 OR title:\4 OR title:\4^^^^,

PRESERVED_PLACEHOLDER_^^^^4all:\4 OR title:\4 OR abs:\4^^^^

These complexes are Alexander dual to total-PRESERVED_PLACEHOLDER_^^^^4all:\4 OR title:\44^^^^-cut complexes (&&&^^^^4all:\4all:\4^^^^&&&).

A fourth usage appears in the strong ring of simplicial complexes, where robustness means rigid algebraic behavior under sums and products: Euler characteristic, Wu characteristics, and the Poincaré polynomial are ring homomorphisms; spectra of the Hodge Laplacian add under products, spectra of the connection Laplacian multiply; and every connected element in the strong ring has a unique prime factorization (&&&^^^^4all:\4 OR title:\4^^^^&&&).

In this broader landscape, the strongly robust simplicial complex PRESERVED_PLACEHOLDER_^^^^4all:\4 OR title:\4^^^^5 is best understood as a specialized algebraic-combinatorial invariant of simple toric ideals. Its faces classify exactly which bouquet patterns preserve strong robustness, while neighboring simplicial-complex literatures use “robustness” for extension properties, collapse invariance, or anti-sparsity conditions rather than for PRESERVED_PLACEHOLDER_^^^^4all:\4 OR title:\4^^^^6 itself.