Discrete & Computational Geometry最新文献

For a discrete poset ({mathcal {X}}), McCord proved that the natural map (|{{mathcal {X}}}|rightarrow {{mathcal {X}}}), from the order complex to the poset with the Up topology, is a weak homotopy equivalence. Much later, Živaljević defined the notion of order complex for a topological poset. For a large class of topological posets we prove the analog of McCord’s theorem, namely that the natural map from the order complex to the topological poset with the Up topology is a weak homotopy equivalence. A familiar topological example is the Grassmann poset (mathcal {G}_n(mathbb {{mathbb {R}}})) of proper non-zero linear subspaces of ({mathbb {R}}^{n+1}) partially ordered by inclusion. But our motivation in topological combinatorics is to apply the theorem to posets associated with tropical phased matroids over the tropical phase hyperfield, and in particular to elucidate the tropical version of the MacPhersonian Conjecture. This is explained in Sect. 2.

In this paper, we give a spectral approximation result for the Laplacian on submanifolds of Euclidean spaces with singularities by the (epsilon )-neighborhood graph constructed from random points on the submanifold. Our convergence rate for the eigenvalue of the Laplacian is (Oleft( left( log n/nright) ^{1/(m+2)}right) ), where m and n denote the dimension of the manifold and the sample size, respectively.

For a lattice path (nu ) from the origin to a point (a, b) using steps (E=(1,0)) and (N=(0,1)), we construct an associated flow polytope ({mathcal {F}}_{{widehat{G}}_B(nu )}) arising from an acyclic graph where bidirectional edges are permitted. We show that the flow polytope ({mathcal {F}}_{{widehat{G}}_B(nu )}) admits a subdivision dual to a ((w-1))-simplex, where w is the number of valleys in the path ({overline{nu }} = Enu N). Refinements of this subdivision can be obtained by reductions of a polynomial (P_nu ) in a generalization of Mészáros’ subdivision algebra for acyclic root polytopes where negative roots are allowed. Via an integral equivalence between ({mathcal {F}}_{{widehat{G}}_B(nu )}) and the product of simplices (Delta _atimes Delta _b), we thereby obtain a subdivision algebra for a product of two simplices. As a special case, we give a reduction order for reducing (P_nu ) that yields the cyclic (nu )-Tamari complex of Ceballos, Padrol, and Sarmiento.

Let K be a compact convex set in (mathbb {R}^{2}) and let (mathcal {F}_{1}, mathcal {F}_{2}, mathcal {F}_{3}) be finite families of translates of K such that (A cap B ne emptyset ) for every (A in mathcal {F}_{i}) and (B in mathcal {F}_{j}) with (i ne j). A conjecture by Dol’nikov is that, under these conditions, there is always some (j in { 1,2,3 }) such that (mathcal {F}_{j}) can be pierced by 3 points. In this paper we prove a stronger version of this conjecture when K is a body of constant width or when it is close in Banach-Mazur distance to a disk. We also show that the conjecture is true with 8 piercing points instead of 3. Along the way we prove more general statements both in the plane and in higher dimensions. A related result was given by Martínez-Sandoval, Roldán-Pensado and Rubin. They showed that if (mathcal {F}_{1}, dots , mathcal {F}_{d}) are finite families of convex sets in (mathbb {R}^{d}) such that for every choice of sets (C_{1} in mathcal {F}_{1}, dots , C_{d} in mathcal {F}_{d}) the intersection (bigcap _{i=1}^{d} {C_{i}}) is non-empty, then either there exists (j in { 1,2, dots , n }) such that (mathcal {F}_j) can be pierced by few points or (bigcup _{i=1}^{n} mathcal {F}_{i}) can be crossed by few lines. We give optimal values for the number of piercing points and crossing lines needed when (d=2) and also consider the problem restricted to special families of convex sets.

We prove an equivalence between open questions about the embeddability of the space of persistence diagrams and the space of probability distributions (i.e. Wasserstein space). It is known that for many natural metrics, no coarse embedding of either of these two spaces into Hilbert space exists. Some cases remain open, however. In particular, whether coarse embeddings exist with respect to the p-Wasserstein distance for (1le ple 2) remains an open question for the space of persistence diagrams and for Wasserstein space on the plane. In this paper, we show that embeddability for persistence diagrams is equivalent to embeddability for Wasserstein space on (mathbb {R}^2). When (p > 1), Wasserstein space on (mathbb {R}^2) is snowflake universal (an obstruction to embeddability into any Banach space of non-trivial type) if and only if the space of persistence diagrams is snowflake universal.

In the classical linear degeneracy testing problem, we are given n real numbers and a k-variate linear polynomial F, for some constant k, and have to determine whether there exist k numbers (a_1,ldots ,a_k) from the set such that (F(a_1,ldots ,a_k) = 0). We consider a generalization of this problem in which F is an arbitrary constant-degree polynomial, we are given k sets of n real numbers, and have to determine whether there exists a k-tuple of numbers, one in each set, on which F vanishes. We give the first improvement over the naïve (O^*(n^{k-1})) algorithm for this problem (where the (O^*(cdot )) notation omits subpolynomial factors). We show that the problem can be solved in time (O^*left( n^{k - 2 + frac{4}{k+2}}right) ) for even k and in time (O^*left( n^{k - 2 + frac{4k-8}{k^2-5}}right) ) for odd k in the real RAM model of computation. We also prove that for (k=4), the problem can be solved in time (O^*(n^{2.625})) in the algebraic decision tree model, and for (k=5) it can be solved in time (O^*(n^{3.56})) in the same model, both improving on the above uniform bounds. All our results rely on an algebraic generalization of the standard meet-in-the-middle algorithm for k-SUM, powered by recent algorithmic advances in the polynomial method for semi-algebraic range searching. In fact, our main technical result is much more broadly applicable, as it provides a general tool for detecting incidences and other interactions between points and algebraic surfaces in any dimension. In particular, it yields an efficient algorithm for a general, algebraic version of Hopcroft’s point-line incidence detection problem in any dimension.

We study the limits of the cones of symmetric nonnegative polynomials and symmetric sums of squares, when expressed in power-mean or monomial-mean basis. These limits correspond to forms with stable expression in power-mean polynomials that are globally nonnegative (resp. sums of squares) regardless of the number of variables. We introduce partial symmetry reduction to describe the limit cone of symmetric sums of squares, and reprove a result of Blekherman and Riener (Discrete Comput Geom 65:1–36, 2020) that limits of symmetric nonnegative polynomials and sums of squares agree in degree 4. We use tropicalization of the dual cones, first considered in the context of comparing nonnegative polynomials and sums of squares in Blekherman et al. (Trans Am Math Soc 375(09):6281–6310, 2022), to show differences between cones of symmetric polynomials and sums of squares starting in degree 6, which disproves a conjecture of Blekherman and Riener (Discrete Comput Geom 65:1–36, 2020). For even symmetric nonnegative forms and sums of squares we show that the cones agree up to degree 8, and are different starting with degree 10. We also find, via tropicalization, explicit examples of symmetric forms that are nonnegative but not sums of squares in the limit.

Delone sets are discrete point sets X in ({mathbb {R}}^d) characterized by parameters (r, R), where (usually) 2r is the smallest inter-point distance of X, and R is the radius of a largest “empty ball” that can be inserted into the interstices of X. The regularity radius ({hat{rho }}_d) is defined as the smallest positive number (rho ) such that each Delone set with congruent clusters of radius (rho ) is a regular system, that is, a point orbit under a crystallographic group. We discuss two conjectures on the growth behavior of the regularity radius. Our “Weak Conjecture” states that ({hat{rho }}_{d}={textrm{O}(d^2log _2 d)}R) as (drightarrow infty ), independent of r. This is verified in the paper for two important subfamilies of Delone sets: those with full-dimensional clusters of radius 2r and those with full-dimensional sets of d-reachable points. We also offer support for the plausibility of a “Strong Conjecture”, stating that ({hat{rho }}_{d}={textrm{O}(dlog _2 d)}R) as (drightarrow infty ), independent of r.

The Sylvester–Gallai Theorem states that every rank-3 real-representable matroid has a two-point line. We prove that, for each (kge 2), every complex-representable matroid with rank at least (4^{k-1}) has a rank-k flat with exactly k points. For (k=2), this is a well-known result due to Kelly, which we use in our proof. A similar result was proved earlier by Barak, Dvir, Wigderson, and Yehudayoff and later refined by Dvir, Saraf, and Wigderson, but we get slightly better bounds with a more elementary proof.

It is broadly known that any parallelepiped tiles space by translating copies of itself along its edges. In earlier work relating to higher-dimensional sandpile groups, the second author discovered a novel construction which fragments the parallelepiped into a collection of smaller tiles. These tiles fill space with the same symmetry as the larger parallelepiped. Their volumes are equal to the components of the multi-row Laplace determinant expansion, so this construction only works when all of these signs are non-negative (or non-positive). In this work, we extend the construction to work for all parallelepipeds, without requiring the non-negative condition. This naturally gives tiles with negative volume, which we understand to mean canceling out tiles with positive volume. In fact, with this cancellation, we prove that every point in space is contained in exactly one more tile with positive volume than tile with negative volume. This is a natural definition for a signed tiling. Our main technique is to show that the net number of signed tiles doesn’t change as a point moves through space. This is a relatively indirect proof method, and the underlying structure of these tilings remains mysterious.