Graph-Based Blob Zeros

• Graph-Based Blob Zeros are structural zeros in functions or matrices organized by graphs, serving as critical invariants in combinatorial, geometric, and analytic studies.
• They underpin factorization in copositive matrices, control hidden zeros in physical amplitudes, and signal phase transitions in statistical models.
• Their analysis facilitates novel algorithms for sparse matrix decomposition, convex hull identification, and intrinsic feature detection in image processing.

Graph-based blob zeros denote a set of phenomena in modern mathematics and physics where the structural zeros of functions or matrices—often organized or interpreted using graphs—underpin the analytic, geometric, or combinatorial behavior of the object under study. In multiple subfields, such as copositive matrix theory, quantum field and statistical physics, probability, and image analysis, these zeros manifest as zero loci or regions (“blobs”) tied to combinatorial or geometric data of an underlying graph. They crucially inform factorization, uniqueness, convex-analytic structure, or asymptotic behavior across these domains.

1. Zeros of Copositive Matrices and Maximal Cliques

For a symmetric copositive matrix $A\in \mathbb{R}^{n\times n}$, the set of normalized zeros,

$$T(A) = \{ z \in \mathbb{R}^n : z \geq 0,\, \|z\|_1 = 1,\, z^T A z = 0 \},$$

admits a canonical decomposition based on graph theory. The procedure centers on identifying minimal zeros—those whose support cannot be further reduced—leading to a finite set $\{\tau(j)\}_{j\in J}$. By constructing the minimal-zeros graph $G(A)$, where nodes are minimal zeros and edges connect pairs $(i,j)$ whenever $\tau(i)^T A \tau(j)=0$, $T(A)$ is represented as

$$T(A) = \bigcup_{K\in \mathcal{K}(G(A))} \mathrm{conv}\{\tau(j):\, j\in K\},$$

where $\mathcal{K}(G(A))$ denotes the maximal cliques and the union is over their convex hulls. This is a unique polyhedral decomposition with minimal cardinality, and its construction encompasses exponential-time enumeration of supports, graph building, clique extraction, and convex hull formation. The omnipresence of such “blob zero” sets manifests in explicit cases, such as perturbed Horn matrices, demonstrating universality of the blob-convex-hull paradigm in copositive analysis (Kostyukova et al., 2024).

2. Graph-based Hidden Zeros in Physical Amplitudes and Wavefunctions

In the context of cosmological wavefunctions and scattering amplitudes, graph-based blob zeros (“hidden zeros”) generalize classical factorization concepts. Given a generating graph $G$ encoding the amplitude object $$T(A) = \{ z \in \mathbb{R}^n : z \geq 0,\, \|z\|_1 = 1,\, z^T A z = 0 \},$$0, cutting $$T(A) = \{ z \in \mathbb{R}^n : z \geq 0,\, \|z\|_1 = 1,\, z^T A z = 0 \},$$1 by boundary vertices $$T(A) = \{ z \in \mathbb{R}^n : z \geq 0,\, \|z\|_1 = 1,\, z^T A z = 0 \},$$2 partitions it into subgraphs $$T(A) = \{ z \in \mathbb{R}^n : z \geq 0,\, \|z\|_1 = 1,\, z^T A z = 0 \},$$3. Imposing the vanishing of all pairwise kinematic invariants $$T(A) = \{ z \in \mathbb{R}^n : z \geq 0,\, \|z\|_1 = 1,\, z^T A z = 0 \},$$4 induces a “blob zero.” The critical result is the dual shuffle factorization: such a blob-zero is equivalent to the amplitude-level shuffle product decomposition

$$T(A) = \{ z \in \mathbb{R}^n : z \geq 0,\, \|z\|_1 = 1,\, z^T A z = 0 \},$$5

where $$T(A) = \{ z \in \mathbb{R}^n : z \geq 0,\, \|z\|_1 = 1,\, z^T A z = 0 \},$$6 denotes summing over all order-preserving interleavings of the external legs of $$T(A) = \{ z \in \mathbb{R}^n : z \geq 0,\, \|z\|_1 = 1,\, z^T A z = 0 \},$$7 and $$T(A) = \{ z \in \mathbb{R}^n : z \geq 0,\, \|z\|_1 = 1,\, z^T A z = 0 \},$$8. Near-blob-zero loci interpolate to conventional pole factorization. This zero structure underpins the uniqueness of tree-level cosmological amplitudes solely via locality and zeros, without recourse to unitarity, and it directly controls enhanced large-$$T(A) = \{ z \in \mathbb{R}^n : z \geq 0,\, \|z\|_1 = 1,\, z^T A z = 0 \},$$9 scaling in BCFW shift analyses (Li et al., 1 Apr 2026).

These blob-zero loci unify and extend the well-structured world of parametric zeros, chain-graph zeros, and more intricate branch and tube cuts, forming the combinatorial skeleton for both physical and mathematical factorization principles.

3. Blob Zeros in Graph-Counting Polynomials and Statistical Physics

For graph-counting polynomials, such as the matching (monomer-dimer) polynomial or partition functions of spin models, the locus of zeros in the complex plane (Lee-Yang or Fisher zeros) forms a “blob” whose analytic structure governs macroscopic statistical behavior. Formally, given a generating function $\{\tau(j)\}_{j\in J}$0 indexed over subgraphs or configurations, the asymptotic distribution of its zeros determines central limit and local limit properties for naturally associated random variables. If the blob of zeros avoids the region of physical fugacity, Gaussian fluctuations prevail; pinchings of the blob (i.e., zeros approaching the positive real axis) signal phase transitions or critical behavior (Lebowitz et al., 2014).

In the Ising model, for a graph $\{\tau(j)\}_{j\in J}$1 of maximal degree $\{\tau(j)\}_{j\in J}$2, the region

$\{\tau(j)\}_{j\in J}$3

is guaranteed to be zero-free for the partition function; this zero-free “blob” is both algorithmically and physically optimal under complexity assumptions (Patel et al., 2023). Blob zeros, in this context, have direct implications for computational tractability and phase structure.

4. Blob Zeros and Matrix Factorization in Graphical Models

Structured zeros (“blobs”) in matrices indexed by a graph are key to sparse matrix factorization, particularly in graphical models. For positive definite matrices $\{\tau(j)\}_{j\in J}$4 respecting the sparsity dictated by a graph $\{\tau(j)\}_{j\in J}$5, preservation of zeros through Cholesky factorization (and its inverse) occurs if and only if $\{\tau(j)\}_{j\in J}$6 is “co-chordal” (homogeneous) and the vertex ordering respects the Hasse-tree decomposition. More formally, for labelling $\{\tau(j)\}_{j\in J}$7, the equivalence

$\{\tau(j)\}_{j\in J}$8

holds if and only if $\{\tau(j)\}_{j\in J}$9 matches these graphical constraints (Khare et al., 2011). Determinants of submatrices indexed by maximal cliques directly encode the blob-zero propagation, supporting their probabilistic interpretations in Gaussian Markov random fields.

5. Blob Zeros in Riemannian Image Analysis

In Riemannian blob detection, the “blob zeros” are defined as zero or extremal sets of curvature-based response functions on the image graph—a submanifold embedded in the product domain and value space. For a scale-space image $G(A)$0, the principal curvatures $G(A)$1 of the graph generate scalar functions $G(A)$2 and $G(A)$3. Blob boundaries are traced by the zero-level sets of $G(A)$4; blob centers correspond to spacetime extrema of the normalized function $G(A)$5. These zeros generalize classical grayscale blob detection methods to manifold-valued images and possess coordinate and channel mixing invariance; they also provide intrinsic feature detection on manifolds (Shestov et al., 2019).

6. Geometric and Polyhedral Interpretations

In both amplitude theory and combinatorics, the zero loci of graph-based objects naturally admit polytopal (associahedral) interpretations. For example, the stripped wavefunction coefficients of chain graphs, when re-expressed in “tube” variables, correspond to hyperplanes $G(A)$6 truncating a simplex, with the structure of the resulting graph associahedron directly encoding the collection of “blob zeros” (De et al., 30 Mar 2025). The adjacency and geometry of these zeros control factorization, zero loci, and piecewise convexity of the associated forms.

7. Physical and Mathematical Significance

Graph-based blob zeros function as fundamental combinatorial and geometric invariants. In copositive matrices, they uniquely encode the convex structure of the zero set. In physics, they provide the underlying reason for factorization properties and uniqueness of amplitudes and wavefunctions and are connected to the deep analytic structure of partition functions. In matrix analysis, they ensure the propagation of sparsity and interpretable independence structure across decompositions. In discrete geometry or image processing, they afford robust, intrinsic means to detect structure by leveraging zero patterns in curvature-induced response functions. The universality and analytic tractability of blob-zero phenomena suggest ongoing importance across algebra, combinatorics, statistical mechanics, quantum field theory, and computational geometry.