We describe the structure of simplicial locally convex fans associated to even-dimensional complete toric varieties with signature 0. They belong to the set of such toric varieties whose even degree Betti numbers yield top gamma vector components equal to 0. The gamma vector is an invariant of palindromic polynomials whose nonnegativity lies between unimodality and real-rootedness. It is expected that the cases where the top components are 0 are among the "building blocks'' of those where it is nonnegative. This means minimality with respect to a certain restricted class of blowups. However, this equality to 0 case is currently poorly understood. In the course of addressing this situation for such toric varieties, we find that the interpretation of the top component of the gamma vector as a signature encodes *intrinsic* combinatorial information on the fan in addition to compatibility with existing natural combinatorial examples as shown earlier.
Our main method involves wall crossings. The links of the fan come from a repeated suspension of the maximal linear subspace in its realization in the ambient space of the fan. Conversely, the centers of these links containing any particular line form a cone or a repeated suspension of one. The intersection patterns between these "anchoring'' linear subspaces come from how far certain submodularity inequalities are from equality and parity conditions on their dimensions. This involves linear dependence and containment relations between them and has connections to optimization. We obtain these relations by viewing the vanishing of certain mixed volumes from the perspective of the exponents. Finally, these wall crossings yield a simple method of generating induced 4-cycles expected to cover the minimal objects described above. Note that this involves intersections of rational equivalence relations with 2-dimensional orbit closures instead of 1-dimensional ones as in most combinatorial applications.
The main decomposition comes from considering doubly Cohen-Macaulay simplicial complexes such that removing a vertex does not change the dimension. In this setting, the coefficients of the h-polynomial are bounded below by those of the double suspension of the link over any edge of the starting simplicial complex. In other words, the remainder has nonnegative coefficients. Note that this is also true if we replace this double suspension by the boundary of a cross polytope of the same dimension by work of Athanasiadis. If we start with a flag sphere, we can “approximate” its h-polynomial by a double suspension of its link over an edge (which itself is a codimension 2 flag sphere) and get to the original flag sphere by a “net nonnegative” number of edge subdivisions. In other words, the second step involves a sequence of edge subdivisions and contractions where there are at least as many edge subdivisions as contractions.
Using this procedure, we can obtain the gamma vector of the flag sphere after taking repeated approximations of the resulting terms by double suspensions of links over edges and taking remainders from a net nonnegative number of edge subdivisions. The vertices that are Boolean (in the sense of a decomposition proposed by Nevo-Petersen-Tenner for simplicial complexes whose f-vectors are equal to the h-vector of a given flag sphere) are from the boundary of a cross polytope and the non-Boolean vertices are those that track remainders from approximations by double suspensions of links over edges. At every step, we have a sum of terms which are products of basis elements of reciprocal/palindromic polynomials of a given dimension and h-polynomials over links over m disjoint edges and points at step m. Note that the degrees involved in the double suspension approximation and remainder parts are formally similar to degrees coming up in geometric Lefschetz decompositions. In particular, we show that remainders of the double suspension approximations are closely related to analogues of the Lefschetz maps which would appear in such a decomposition and their images after an identification involving the Boolean vertices. This involves how the remainders behave under edge subdivisions/contractions and tracing how explicit vertices in the decomposition are involved in spanning the Artinian reduction of the Stanley-Reisner ring of the given flag sphere. Looking towards local vs. global Boolean decompositions (taking “local” to mean links over edges), this indicates that having a change in at most 1 locally Boolean vertex to a non-Boolean one would mean that it extends to a globally Boolean one.
We consider Lefschetz-type decompositions of simplicial complexes whose f-vectors are equal to h-vectors of classes of simplicial complexes closed under (double) suspensions. For the latter, we take those of a given dimension to be PL homeomorphic to each other. When the “primitive” part is Boolean and the simplicial complexes in the class are flag, whether we can patch together such a decomposition from certain codimension 2 local parts depends on if a certain map formally satisfies properties of the Hard Lefschetz theorem. When we consider h-vectors of simplicial pseudomanifolds, these local parts are codimension 2 flag spheres. The map (on the simplicial complexes we take h-vectors of) is a composition of a double suspensions and a “net single edge subdivision”. Finally, the fact that the simplicial complexes we take f-vectors of can be assumed to be balanced suggests possible algebraic versions of these maps.
S. Park, f-vectors of balanced simplicial complexes, flag spheres, and geometric Lefschetz decompositions
The use of the Hard Lefschetz theorem has had a long history of being involved in unimodality results in algebraic combinatorics (e.g. h-vectors of boundaries of simplicial polytopes in work of Stanley). For the most part, they are related to the h-vector, which is naturally related to algebraic structures involving the given simplicial complex. Here, we find f-vectors of (d – 1)-dimensional simplicial complexes with d-colorable 1-skeleta that seem to satisfy a Lefschetz decomposition-like structure with Boolean lattices playing the role of primitive parts. This builds on work of Nevo-Petersen and Nevo-Petersen-Tenner on gamma vectors of odd-dimensional flag spheres. Note that this has some resemblance to a construction of Björner-Frankl-Stanley of balanced shellable simplicial complexes whose h-vectors are equal to f-vectors of with d-colorable 1-skeleta involving “filling in” colors that are missing from individual faces of the starting simplicial complex.
Given a flag sphere, it is known that its h-vector is equal to the f-vector of a simplicial complex with a d-colorable 1-skeleton. We use the unique facet of the latter simplicial complex as a “color palette” for the simplicial complex for the vertex set yielding the Boolean part of the decomposition. This is based on thinking about the 1-dimensional example. The inductive step/local-global argument is based on a partition of unity lemma from Adiprasito’s proof of the g-conjecture by (d – 3)-dimensional subcomplexes whose f-vector is equal to the h-vector of codimension 2 flag spheres covering the original one with the same coloring interpretation. The Dehn-Sommerville relations are involved in the Boolean part of the (d – 2)-dimensional faces. Note that the enumerative consequences of a Boolean decomposition (e.g. what the gamma vector is) can be shown using a *correspondence* with *pairs* of independent components (P, Q) where Q can be any element of the Boolean lattice of a certain size depending on |P|. The objects P and Q do not necessarily have to have the same type of object. We can apply our work to positivity questions related to reciprocal/palindromic polynomials associated to flag spheres. Finally, we note that the Lefschetz decomposition we study uses the same degrees as the geometric Lefschetz decomposition for compact Kähler manifolds instead of halved degrees in the usual h-vector setting.
One of the most common and effective methods of obtaining structural information on simplicial complexes is to use tools from algebraic geometry/commutative algebra (often motivated by properties of toric varieties). However, there is no general algebro-geometric description of components of the gamma vector holding for arbitary flag simplicial spheres. This invariant occurs in many different contexts including permutation statistics, signatures of toric varieties, and Euler characteristics of nonpositively curved piecewise Euclidean manifolds. Combinatorial methods resulting from an explicit inverted Chebyshev expansion give rise to new positivity properties and cell complex structures that are of interest in their own right. Note that the focus is on the f-vector rather than the h-vector in "algebraic" settings. For flag simplicial spheres Delta, the fact that h(Delta) = f(Gamma) and compatibility between Chebyshev expansions and a modification of the f-polynomial by work of Hetyei are the key inputs. In the main formula implying new positivity results, local structures of CAT(0) complexes and cubical analogues of barycentric subdivisions give deeper connections with cubical complex structures complementing earlier work related to the top gamma vector component. Afterwards, we return to the motivating example of barycentric subdivisions and consider how f-vectors of Cohen--Macaulay and vertex decomposable flag complexes in geometric settings decompose and interact with geometric transformations. This includes subdivisions of simplicial complexes and recursive properties they share with vertex decomposable flag complexes.
Given a reciprocal/palindromic polynomial of even degree, we show that the gamma vector is essentially given by an inverted Chebyshev polynomial basis expansion. As an immediate consequence, we characterize real-rootedness of a linear combination of Chebyshev polynomials in terms of real-rootedness of that of the reciprocal polynomial built out of an inverted scaled tuple of the coefficients with one fixed and the rest divided by 2. It can be taken as a counterpart for arbitrary dimensions of a recent result of Bel-Afia-Meroni-Telen on hyperbolicity of Chebyshev curves with respect to the origin. In general, Chebyshev varieties serve as a counterpart of toric varieties in sparse polynomial root finding. Apart from this, the inverted Chebyshev expansion also yields connections between intrinsic properties of the gamma vector construction and the geometric combinatorics of simplicial complexes and posets.
We find this by applying work of Hetyei on Tchebyshev subdivisions and Tchebyshev posets. In particular, we find that the gamma vector transformation is closely related to f-vectors of simplicial complexes resulting from successive edge subdivisions that transform the type A Coxeter complex to the type B Coxeter complex. Lifting to this to the level of ce-indices (a modification of cd-indices), we show that the gamma vector inverted Chebyshev polynomial expansion lifts to a sum of ce-indices of cross polytope triangulations which can be computed using descent statistics involving edge labelings of maximal chains. While there are many examples in the literature where gamma positivity involving descent statistics, it is interesting to note that we have this without initial structural assumptions on the input polynomial apart from being reciprocal/palindromic. Finally, we note that all of these can be repeated with Chebyshev polynomials of the second kind after taking derivatives. This gives connections to Hopf algebras and quasisymmetric functions along with Lefschetz-type maps.
We give explicit formulas for the gamma vector directly in terms of the input polynomial which extends to arbitrary polynomials. More specifically, we can express them as a linear combination of the coefficients of the starting polynomial (using Catalan numbers and binomial coefficients) or in terms of the derivative of a simple quotient of the input polynomial. The first formula suggests relations to common noncrossing partition/Coxeter group structures in existing gamma positivty examples. The second one shows that the gamma vector measures the difference between local and global information when the input polynomial is the h-polynomial of a simplicial complex. It can also be used to relate signs/inequalities of the gamma vector to upper/lower bounds for the coefficients of the input polynomial. The expression in terms of a derivative and relation to the bounds complements and extends initial observations made by Gal when he defined the gamma vector. Finally, we use the form taken by sums involved while making these estimates and connections between matroid-related examples and intersection numbers to connect these properties of the gamma vector to algebraic structures. For example, this includes characteristic classes involved in log concavity and Schur positivity problems.
S. Park, Symmetries and intrinsic vs. extrinsic properties of M_{0, n}bar
Our motivation is the following question: How much of the combinatorial structure of the moduli space of stable rational curves with n marked points is “intrinsic” to (the geometry of) the space itself? We view this from the lens of the natural permutation action on the points. In fact, it is known that this action does *not* extend to other wonderful compactifications of the complement of the A_{n – 2} hyperplane arrangement. We determine the differences in intersection patterns of parallel faces of associahedra and permutohedra inducing this rigidity property. As a consequence, we show that this is reflected in most of terms of degree at least 2 in the Chow ring. On the other hand, we consider degree 1 elements from the perspective of S_n-invariance (e.g. suitable Lefschetz elements) and how it connects to recent positivity results. In particular, we show that the log concave sequences arising from degree 1 Hodge-Riemann relations using S_n-invariant elements of the Picard group have a special (recursive) structure. They take the form of polynomials in (quantum) Littlewood-Richardson coefficients multiplied by terms such as partition components, factorials, and multinomial coefficients. Finally, we connect higher degree Hodge-Riemann relations (of other rings) to the geometry of this moduli space via Toeplitz matrices.
S. Park, Matroidal Cayley-Bacharach and independence/dependence of geometric properties of matroids
We study the relationship between “geometric” properties of matroids and the matroidal Cayley-Bacharach property of degree a MCB(a) defined by Levinson and Ullery. From the perspective of matroid polytopes/generalized permutohedra, we see from the case of nestohedra that the MCB(a) property can have a natural description in terms of properties of polytopes while *not* being a combinatorial invariant of polytopes. On the other hand, there seem to be a close relationship between combinatorial properties in the case of paving matroids (which are conjecturally almost all matroids of a given rank) and supersolvable line/hyperplane arrangements. The paving matroids involve a relationship between the degree a and the Chow rings of matroids. Using supersolvable line arrangements, we find a family of matroids other than the case of representable matroids where the MCB(a) property measures the failure of a set of points to impose independent conditions on the space of hypersurface of a given degree. In general, MCB(a) for supersolvable hyperplane arrangements has a recursive property from MCB(b) for lower degrees b and covers of appropriate subarrangements.
S. Park, Anti-Ramsey theory problems, lattice point counts on polytopes, and Hodge structures on the cohomology of toric varieties
By "anti--Ramsey theory problems", we mean the number of edge colorings of graphs such that a specified subgraphs are *not* monochromatic. We find families of graphs and subgraphs such that this number is determined by a lattice point count. The idea is to combine a reinterpretation of simplicial chromatic polynomials and connections between h-vectors and lattice point counts of polytopes. Note that this follows up on our earlier work expressing simplicial chromatic polynomials in terms of h-vectors of auxiliary simplicial complexes. As a result, we obtain a family of “anti-Ramsey” questions addressed using geometric/structural methods.
S. Park, Matroids satisfying the matroidal Cayley--Bacharach property and ranks of covering flats
We first show that there are no nontrivial bounds on ranks of proper flats that cover the underlying set of a matroid satisfying the matroidal analogue of the Cayley-Bacharach property. This gives a negative answer to a question in recent work of Levinson and Ullery. Next, we look at the matroidal Cayley-Bacharach property from the point of view of polytopes associated with matroids that are studied. We consider a (generic) class of matroids where the matroidal Cayley-Bacharach property depends on a collection of set-theoretic properties depending on the ranks of the flats of the matroids arising from the polytopes.
S. Park, Simplicial chromatic polynomials as Hilbert series of Stanley--Reisner rings
This project started with our initial observation that Euler characteristic-like invariants of ordered configuration spaces of distinct points on a manifold and can be altered to obtain chromatic polynomials of graphs. This means only preventing the coordinates corresponding to adjacent vertices from being equal to each other. As it turns out, properties of this modified configuration space such as this one were studied earlier by Eastwood and Huggett and there is a higher-dimensional version of this connection arising from simplicial complexes in work of Cooper-de Silva-Sazdanovic. This polynomial (the simplicial chromatic polynomial) is uniquely determined up to normalization by a deletion-contraction type relation. While they study the polynomial from a topological point of view, we find an explicit combinatorial interpretation for a large class of initial simplicial complexes. More specifically, we find that they arise from the Hilbert series of Stanley-Reisner rings associated to auxiliary simplicial complexes. In addition, the minimal nonfaces of *any* simplicial complex used as input for the simplicial chromatic polynomial can be those of the auxiliary simplicial complex of *some* simplicial complex used for the h-vector.
Note that these polynomials are closely related to characteristic polynomials of diagonal/hypergraph linear subspace arrangements (or their associated polymatroids). Since they are determined by h-vectors of auxiliary simplicial complexes, we found some connections between these simplicial chromatic polynomials and other questions involving log concavity, symmetries between a polynomial and its reciprocal polynomial, and cyclotomic polynomials along the way.
S. Park, Graph coloring-related properties of (generating functions of) Hodge--Deligne polynomials
We were considering some connections between Euler characteristic-like invariants (e.g. Hodge-Deligne polynomials) of configuration spaces and chromatic polynomials. It turns out that there is a connection between colorings of *directed* graphs and Hodge-Deligne polynomials as well. We take a look at what this means and how it relates to existing structures between Hodge numbers (e.g. birational invariants) and properties of configuration spaces.
S. Park, Characterizing cubic hypersurfaces via projective geometry
Under certain numerical/generic conditions, we show that cubic hypersurfaces are characterized by a projective geometry construction. This uses a cut and paste relation (in the Grothendieck ring of varieties) of Galkin and Shinder matching pairs of points with an incidence correspondence involving the third point of intersection and the line spanned by the first two points (filtering out instances where the line is contained in the given variety). Weakening these conditions extends the possibilities to complete intersections of two quadric hypersurfaces or two quartic hypersurfaces. As a special case, we find generic hypersurfaces of a given degree satisfying this cut and paste relation must be cubic hypersurfaces.
S. Park, Motivic limits for Fano varieties of k-planes, The Quarterly Journal of Mathematics, haac012 (2022)
We show that "most'' of certain properties of Fano varieties of k-planes (k-planes contained in a given projective variety) are determined by symmetric products of points on the given variety, Grassmannians of appropriate dimensions, and incidence correspondences of points in linear subspaces. Examples of properties in question are those compatible with cut and paste constructions such as Poincare polynomials, Euler characteristics, and Hodge-Deligne polynomials. The main idea is to construct an approximate/motivic limit version of a relation of Galkin and Shinder in the Grothendieck ring of varieties. This means building a correspondence between points and incidence correspondences coming from the intersection of a variety in projective space with a linear subspace of complementary dimension and filtering out loci where this map is not a bijection (which includes terms from Fano varieties of k-planes).
S. Park, Decomposability and Mordell-Weil ranks of Jacobians using Picard numbers
We study number field analogues of some questions of Ekedahl and Serre about the decomposability of Jacobians of curves C over number fields as a product of elliptic curves. The main case considered involves self-products E^g and we approach this question by studying the Picard numbers of self-products of the curves C involved under specialization to primes. This involves methods previously used by Costa, Elsenhans, and Jahnel to study those of K3 surfaces. As a result, we give bounds on the genus of such curves with respect to initial arithmetic invariants (e.g. norms of primes related to reduction properties or heights) and obtain infinite families where the reduction modulo a prime is maximal or minimal when such decompositions exist. In addition, we rule out cases where the curves have a large automorphism group. Finally, we show that Picard numbers of self-products of curves can also be used to study jumps of Mordell-Weil ranks via results of Ulmer on Mordell-Weil ranks of Jacobians over function fields and endomorphism rings.
L. Chua, B. Gunby, S. Park, and A. Yuan, Proof of a conjecture of Guy on class numbers, International Journal of Number Theory, 11 (2015), pages 1345 - 1355.
We resolve a conjecture of Guy on a congruence between class numbers of quadratic fields Q(sqrt(\pm p)) and continued fraction expansions of \sqrt(p). The tools used were some algebraic number theory, results of Zagier connecting these class numbers with the continued fraction expansions, Jacobi symbols, and Dedekind sums. While this question is apparently about class numbers, it is interesting to note that the main ideas used are combinatorial arguments rather than the structure of the class group.
L. Chua, S. Park, and G. Smith, Bounded gaps between primes in special sequences, Proceedings of the American Mathematical Society, 143 (2015), pages 4597 - 4611.
In Maynard’s work on bounded gaps between primes, it was shown that any subset of the primes which is “well-distributed” in arithmetic progressions contains many primes which are “close together”. We adapt his method to show that there are bounded gaps between sequences of the form [bn], where b is an irrational number of finite type.
S. Park, Arithmetic properties of generalized Fibonacci sequences
We consider a generalization of the Fibonacci sequence which shares some arithmetic properties with the original sequence. This includes a resolution to some conjectures of Chen, Moll, and Sagan on periodicity, d-adic valuations, and the behavior of an analogue of the Riemann zeta function. Also, we give an algebraic description of the periodicity property considered and study how it is distributed.
S. Park, Discriminators of quadratic polynomials
For polynomials f and positive integers n, we study the discriminator D_f(n), which is the smallest number m such that. f(1),…,f(n) are distinct mod m. This was first defined in the context of computing square roots of a long sequence of numbers for a computer simulation. While this quantity has been studied for certain classes of polynomials, it is very complicated in general. We focus on polynomials of the form f(x) = x(dx – 1) where this problem is more tractable and extend results of Sun for d = 2, 3 where D_f(n) = d^([log_d n] + 1) to d = 2^r for positive integers r. Afterwards, we also study cases where d = p^r for other primes p (e.g. using bounds) and observe using computational methods that discriminator values are concentrated around prime powers even after increasing the size of the prime p or power r. This gives a potential method for generating prime numbers using discriminators of polynomials.