Linear syzygies and birational combinatorics

Let F be a finite set of monomials of the same degree d ≥ 2 in a polynomial ring R = k[x ₁,…, x _n ] over an arbitrary field k. We give some necessary and/or sufficient conditions for the birationality of the ring extension k[F] ? R ^(d), where R ^(d) is the dth Veronese subring of R. One of our results extends to arbitrary characteristic, in the case of rational monomial maps, a previous syzygytheoretic birationality criterion in characteristic zero obtained in [1].     

Elementary symmetric polynomials in Shamir's scheme

The concept of k-admissible tracks in Shamir's secret sharing scheme over a finite field was introduced by Schinzel et al. (2009) [10]. Using some estimates for the elementary symmetric polynomials, we show that the track (1,…,n) over Fp is practically always k-admissible; i.e., the scheme allows to place the secret as an arbitrary coefficient of its generic polynomial even for relatively small p. Here k is the threshold and n the number of shareholders.     

Products of idempotent matrices

For some years it has been known that every singular square matrix over an arbitrary field F is a product of idempotent matrices over F. This paper quantifies that result to some extent. Main result: for every field F and every pair (n,k) of positive integers, an n×n matrix S over F is a product of k idempotent matrices over F iff rank(I ? S)?k· nullity S. The proof of the “if” part involves only elementary matrix operations and may thus be regarded as constructive. Corollary: (for every field F and every positive integer n) each singular n×n matrix over F is a product of n idempotent matrices over F, and there is a singular n×n matrix over F which is not a product of n ? 1 idempotent matrices.     

Blocking sets, k-arcs and nets of order ten

Our first main objective here is to unify two important theories in finite geometries, namely, the theories of k-arcs and blocking sets. This has a number of consequences, which we develop elsewhere. However, one consequence that we do discuss here is an improvement of Bruck's bound [1] concerning the possibility of embedment of finite nets of order n, in the controversial case when n = 10. The argument also makes use of a recent computer result of Denniston [5]. The second (related) main result involves a new combinatorial bound concerning blocking sets (Theorem 5). We are able to show that the bound is sharp by constructing a new class of geometrical examples of blocking sets in Theorem 6. See also the note added in proof.     

On hypergraphs without two edges intersecting in a given number of vertices

Let X be a finite set of n-melements and suppose t ? 0 is an integer. In 1975, P. Erdös asked for the determination of the maximum number of sets in a family $\text{F}$ = {F₁,…, F_m}, F_i ? X, such that ∥F_i ∩ F_j∥ ≠ t for 1 ? i ≠ j ? m. This problem is solved for n ? n₀(t). Let us mention that the case t = 0 is trivial, the answer being 2^{n ? 1}. For t = 1 the problem was solved in [3]. For the proof a result of independent interest (Theorem 1.5) is used, which exhibits connections between linear algebra and extremal set theory.     

Combinatorial Stokes formulas via minimal resolutions

We describe an explicit chain map from the standard resolution to the minimal resolution for the finite cyclic group Zk of order k. We then demonstrate how such a chain map induces a “Zk-combinatorial Stokes theorem,” which in turn implies “Dold's theorem” that there is no equivariant map from an n-connected to an n-dimensional free Zk-complex. Thus we build a combinatorial access road to problems in combinatorics and discrete geometry that have previously been treated with methods from equivariant topology. The special case k=2 for this is classical; it involves Tucker's (1949) combinatorial lemma which implies the Borsuk-Ulam theorem, its proof via chain complexes by Lefschetz (1949), the combinatorial Stokes formula of Fan (1967), and Meunier's work (2006).     

A new short proof of a theorem of Ahlswede and Khachatrian

Ahlswede and Khachatrian [R. Ahlswede, L.H. Khachatrian, The complete nontrivial-intersection theorem for systems of finite sets, J. Combin. Theory Ser. A 76 (1996) 121-138] proved the following theorem, which answered a question of Frankl and Füredi [P. Frankl, Z. Füredi, Nontrivial intersecting families, J. Combin. Theory Ser. A 41 (1986) 150-153]. Let 2?t+1?k?2t+1 and n?(t+1)(k−t+1). Suppose that F is a family of k-subsets of an n-set, every two of which have at least t common elements. If |_?F∈FF|<t, then , and this is best possible. We give a new, short proof of this result. The proof in [R. Ahlswede, L.H. Khachatrian, The complete nontrivial-intersection theorem for systems of finite sets, J. Combin. Theory Ser. A 76 (1996) 121-138] requires the entire machinery of the proof of the complete intersection theorem, while our proof uses only ordinary compression and an earlier result of Wilson [R.M. Wilson, The exact bound in the Erd?s-Ko-Rado theorem, Combinatorica 4 (1984) 247-257].     

On polyvectors of vector spaces and hyperplanes of projective Grassmannians

We investigate relationships between polyvectors of a vector space V, alternating multilinear forms on V, hyperplanes of projective Grassmannians and regular spreads of projective spaces. Suppose V is an n-dimensional vector space over a field F and that A_n-1,k(F) is the Grassmannian of the (k − 1)-dimensional subspaces of PG(V) (1  ? k ? n − 1). With each hyperplane H of A_n-1,k(F), we associate an (n − k)-vector of V (i.e., a vector of ∧^n−kV) which we will call a representative vector of H. One of the problems which we consider is the isomorphism problem of hyperplanes of A_n-1,k(F), i.e., how isomorphism of hyperplanes can be recognized in terms of their representative vectors. Special attention is paid here to the case n = 2k and to those isomorphisms which arise from dualities of PG(V). We also prove that with each regular spread of the projective space PG(2k-1,F), there is associated some class of isomorphic hyperplanes of the Grassmannian A_2k-1,k(F), and we study some properties of these hyperplanes. The above investigations allow us to obtain a new proof for the classification, up to equivalence, of the trivectors of a 6-dimensional vector space over an arbitrary field F, and to obtain a classification, up to isomorphism, of all hyperplanes of A_5,3(F).     

A combinatorial proof of a theorem of Freund

In 1989, Robert W. Freund published an article about generalizations of the Sperner lemma for triangulations of n-dimensional polytopes, when the vertices of the triangulations are labeled with points of Rn. For y∈Rn, the generalizations ensure, under various conditions, that there is at least one simplex containing y in the convex hull of its labels. Moreover, if y is generic, there is generally a parity assertion, which states that there is actually an odd number of such simplices.For one of these generalizations, contrary to the others, neither a combinatorial proof, nor the parity assertion were established. Freund asked whether a corresponding parity assertion could be true and proved combinatorially.The aim of this paper is to give a positive answer, using a technique which can be applied successfully to prove several results of this type in a very simple way. We prove actually a more general version of this theorem. This more general version was published by van der Laan, Talman and Yang in 2001, who proved it in a non-combinatorial way, without the parity assertion.     

Orbits of rational n-sets of projective spaces under the action of the linear group

Let k=Fq be a finite field. We enumerate k-rational n-sets of (unordered) points in a projective space PN over k, and we compute the generating function for the numbers of PGLN+1(k)-orbits of these n-sets. For N=1,2 we obtain a formula for these numbers of orbits as a polynomial in q with integer coefficients.     

A Proof of KÜhnel’s Conjecture for n ⩾ k2 + 3k Eric Sparla

In this note we show that an Upper Bound Conjecture made by Kühnel for combinatorial 2k-manifolds holds for fixed k if its number of vertices is at least n ? k² + 3k. Together with known results this provides a simple proof of the conjecture for k = 1 and k = 2.     

Solving Non-Homogeneous Nested Recursions Using Trees

The solutions to certain nested recursions, such as Conolly’s C(n) = C(n?C(n?1)) + C(n?1?C(n?2)), with initial conditions C(1) = 1, C(2) = 2, have a well-established combinatorial interpretation in terms of counting leaves in an infinite binary tree. This tree-based interpretation, and its generalization to a similar k-term nested recursion, only apply to homogeneous recursions and only solve each recursion for one set of initial conditions determined by the tree. In this paper, we extend the tree-based interpretation to solve a non-homogeneous version of the k-term recursion that includes a constant term. To do so we introduce a tree-grafting methodology that inserts copies of a finite tree into the infinite k-ary tree associated with the solution of the corresponding homogeneous k-term recursion. This technique also solves the given non-homogeneous recursion with various sets of initial conditions.     

On generalised catalan numbers

The Catalan number C_n is defined to be $\text{2n}\text{n}\text{(n+1)}$. One of its occurrences is as the number of ways of bracketing a product of n+1 terms taken from a set with binary operation. In this note the corresponding result for a set with a k-ary operation is considered. A combinatorial proof is given which does not involve generating functions or inversion formulae. The result is further generalised to obtain a simpler proof of a formula of Erdelyi and Etherington [2], interpreted here as a result concerning a set with several k_i-ary operations.     

On a class of similarity solutions of the porous media equation,III

It is shown how existence questions for general multiparameter eigenvalue problems can be treated quite simply using degree theory. The equations to be solved are W_n(λ)x_n = 0 ≠ x_n, n = 1, 2,…, k, where λ ? $\text{R}$^k and each W_n(λ) is a self-adjoint linear operator on a Hilbert space H_n. The W_n, which may be unbounded, depend continuously on λ in a suitable sense. A coercivity condition for large ∥ λ ∥ is used, and is shown to be equivalent, in the “linear” case, to a standard determinantal definiteness condition.     

Large spaces of symmetric matrices of bounded rank are decomposable ∗

Let k and n be positive integers such that kn. Let S_n (F) denote the space of all n×n symmetric matrices over the field F with char F≠2. A subspace L of S_n (F) is said to be a k-subspace if rank Ak for every A?L.

Now suppose that k is even, and write k=2r. We say a k∥-subspace of S_n (F) is decomposable if there exists in Fⁿ a subspace W of dimension n?r such that x^tAx=0 for every x?W A?L.

We show here, under some mild assumptions on k n and F, that every k∥-subspace of S_n (F) of sufficiently large dimension must be decomposable. This is an analogue of a result obtained by Atkinson and Lloyd for corresponding subspaces of F^m,n .     

Forbidden submatrices

We consider a m × n (0, 1)-matrix A, no repeated columns, which has no k × l sumatrix F. We may deduce bounds on n, polynomial in m, depending on F. The best general bound is O(m^2k−1). We improve this and provide best possible bounds for k × 1 F's and certain k × 2 F's. In the case that all columns of F are the same, good bounds are obtained which are best possible for l = 2 and some other cases. Good bounds for 1 × l F's are provided, namely n ⩽ (l−1)m + 1, which are shown to be best possible for F = [1010...10]. The paper finishes with a study of the 14 different 3 × 2 possibilities for F, solving all but 3.     

A representation of convex semilinear sets

If F is an ordered field, a subset of n-space over F is said to be semilinear just in case it is a finite Boolean combination of translates of closed halfspaces, where a closed halfspace is the set of all points obeying a homogeneous weak linear inequality with coefficients from F. Andradas, Rubio, and Vélez have shown that closed (open) convex semilinear sets are finite intersections of translates of closed (open) halfspaces (an open halfspace is defined as before, but with a strict inequality). This paper represents arbitrary convex semilinear sets in a manner analogous to that of Andradas, Rubio, and Vélez.     

A combinatorial proof of a formula for the Lucas-Narayana polynomials

In 2020, Bennett, Carrillo, Machacek and Sagan gave a polynomial generalization of the Narayana numbers and conjectured that these polynomials have positive integer coefficients for $1\le k\le n$ and for $n\ge 1$. In 2020, Sagan and Tirrell used a powerful algebraic method to prove this conjecture (in fact, they extend and prove the conjecture for more than just the type A case). In this paper we give a combinatorial proof of a formula satisfied by the Lucas-Narayana polynomials described by Bennett et al. This gives a combinatorial proof that these polynomials have positive integer coefficients. A corollary of our main result establishes a parallel theorem for the FiboNarayana numbers $N_{n,k,F}$, providing a combinatorial proof of the conjecture that these are positive integers for $n\ge 1$.     

Primitive words,free factors and measure preservation

Let F _k be the free group on k generators. A word w ∈ F _k is called primitive if it belongs to some basis of F _k . We investigate two criteria for primitivity, and consider more generally subgroups of F _k which are free factors. The first criterion is graph-theoretic and uses Stallings core graphs: given subgroups of finite rank H ≤ J ≤ F _k we present a simple procedure to determine whether H is a free factor of J. This yields, in particular, a procedure to determine whether a given element in F _k is primitive. Again let w ∈ F _k and consider the word map w: G × … × G → G (from the direct product of k copies of G to G), where G is an arbitrary finite group. We call w measure preserving if given uniform measure on G × … × G, w induces uniform measure on G (for every finite G). This is the second criterion we investigate: it is not hard to see that primitivity implies measure preservation, and it was conjectured that the two properties are equivalent. Our combinatorial approach to primitivity allows us to make progress on this problem and, in particular, prove the conjecture for k = 2. It was asked whether the primitive elements of F _k form a closed set in the profinite topology of free groups. Our results provide a positive answer for F ₂.