Product Conjecture Research Articles

On the packing/covering conjecture of infinite matroids

The Packing/Covering Conjecture was introduced by Bowler and Carmesin motivated by the Matroid Partition Theorem of Edmonds and Fulkerson. A packing for a family (Mi:i∈Θ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$({M_i}:i \\in \\Theta)$$\\end{document} of matroids on the common edge set E is a system (Si:i∈Θ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$({S_i}:i \\in \\Theta)$$\\end{document} of pairwise disjoint subsets of E where Si is panning in Mi. Similarly, a covering is a system (Ii: i ∈ Θ) with ∪i∈ΘIi=E\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\cup _{i \\in \\Theta}}{I_i} = E$$\\end{document} where Ii is independent in Mi. The conjecture states that for every matroid family on E there is a partition E=Ep⊔Ec\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$E = {E_p} \\sqcup {E_c}$$\\end{document} such that (Mi↾Ep:i∈Θ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$({M_i}\\upharpoonright{E_p}:i \\in \\Theta)$$\\end{document} admits a packing and (Mi.Ec:i∈Θ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$({M_i}.{E_c}:i \\in \\Theta)$$\\end{document} admits a covering. We prove the case where E is countable and each Mi is either finitary or cofinitary. To do so, we give a common generalisation of the singular matroid intersection theorem of Ghaderi and the countable case of the Matroid Intersection Conjecture by Nash-Williams by showing that the conjecture holds for countable matroids having only finitary and cofinitary components.

Intersection of a partitional and a general infinite matroid

Let E be a possibly infinite set and let M and N be matroids defined on E. We say that the pair {M,N} has the Intersection property if M and N share an independent set I admitting a bipartition IM⊔IN such that spanM(IM)∪spanN(IN)=E. The Matroid Intersection Conjecture of Nash-Williams says that every matroid pair has the Intersection property.The conjecture is known and easy to prove in the case when one of the matroids is uniform and it was shown by Bowler and Carmesin that the conjecture is implied by its special case where one of the matroids is a direct sum of uniform matroids, i.e., is a partitional matroid. We show that if M is an arbitrary matroid and N is the direct sum of finitely many uniform matroids, then {M,N} has the Intersection property.

On the intersection conjecture for infinite trees of matroids

Using a new technique, we prove a rich family of special cases of the matroid intersection conjecture. Roughly, we prove the conjecture for pairs of tame matroids which have a common decomposition by 2-separations into finite parts.

Proof of Nash-Williams' intersection conjecture for countable matroids

We prove that if M and N are finitary matroids on a common countable edge set E then they admit a common independent set I such that there is a bipartition E=EM∪EN for which I∩EM spans EM in M and I∩EN spans EN in N. It answers positively the Matroid Intersection Conjecture of Nash-Williams in the countable case.

The Almost Intersection Property for Pairs of Matroids on Common Groundset

We introduce the Almost Intersection Property for pairs of possibly infinite matroids on the same groundset. We prove that if such a pair satisfies the Almost Intersection Property then it satisfies the Matroid Intersection Conjecture of Nash-Williams. We also present some corollaries of that result.

Linear recurrence sequences and the duality defect conjecture

It is conjectured that the dual variety of every smooth nonlinear subvariety of dimension $> \frac{2N}{3}$ in projective $N$-space is a hypersurface, an expectation known as the duality defect conjecture. This would follow from the truth of Hartshorne's complete intersection conjecture but nevertheless remains open for the case of subvarieties of codimension $> 2$. A combinatorial approach to proving the conjecture in the codimension $2$ case was developed by Holme, and following this approach Oaland devised an algorithm for proving the conjecture in the codimension $3$ case for particular $N$. This combinatorial approach gives a potential method of proving the duality defect conjecture in many of the cases by studying the positivity of certain homogeneous integer linear recurrence sequences. We give a generalization of the algorithm of Oaland to the higher codimension cases, obtaining with this bounds the degrees of counterexamples would have to satisfy, and using the relationship with recurrence sequences we prove that the conjecture holds in the codimension $3$ case when $N$ is odd.

On Furstenberg's intersection conjecture, self-similar measures, and the $L^q$ norms of convolutions

We study a class of measures on the real line with a kind of self-similar structure, which we call dynamically driven self-similar measures, and contain proper self-similar measures such as Bernoulli convolutions as special cases. Our main result gives an expression for the $L^q$-dimensions of such dynamically driven self-similar measures, under certain conditions. As an application, we settle Furstenberg's long-standing conjecture on the dimension of the intersections of $\times p$ and $\times q$-invariant sets. Among several other applications, we also show that Bernoulli convolutions have an $L^q$ density for all finite $q$, outside of a zero-dimensional set of exceptions. The proof of the main result is inspired by M. Hochman's approach to the dimensions of self-similar measures and his inverse theorem for entropy. Our method can be seen as an extension of Hochman's theory from entropy to $L^q$ norms, and likewise relies on an inverse theorem for the decay of $L^q$ norms of discrete measures under convolution. This central piece of our approach may be of independent interest, and is an application of well-known methods and results in additive combinatorics: the asymmetric version of the Balog-Szemeredi-Gowers Theorem due to Tao-Vu, and some constructions of Bourgain.

On the unstable intersection conjecture

Compacta X and Y are said to admit a stable intersection in R^n if there are maps f : X -> R^n and g : Y -> R^n such that for every sufficiently close continuous approximations f' : X -> R^n and g' : Y -> R^n of f and g we have f'(X)\cap g'(Y)\neq\emptyset. The well-known conjecture asserting that X and Y do not admit a stable intersection in R^n if and only if dim X \times Y \leq n-1 was confirmed in many cases. In this paper we prove this conjecture in all the remaining cases except the case dim X =dim Y =3, dim X \times Y=4 and n=5 which still remains open.

On the intersection of infinite matroids

We show that the infinite matroid intersection conjecture of Nash-Williams implies the infinite Menger theorem proved by Aharoni and Berger in 2009.We prove that this conjecture is true whenever one matroid is nearly finitary and the second is the dual of a nearly finitary matroid, where the nearly finitary matroids form a superclass of the finitary matroids.In particular, this proves the infinite matroid intersection conjecture for finite-cycle matroids of 2-connected, locally finite graphs with only a finite number of vertex-disjoint rays.

Refined open intersection numbers and the Kontsevich-Penner matrix model

A study of the intersection theory on the moduli space of Riemann surfaces with boundary was recently initiated in a work of R. Pandharipande, J.P. Solomon and the third author, where they introduced open intersection numbers in genus 0. Their construction was later generalized to all genera by J.P. Solomon and the third author. In this paper we consider a refinement of the open intersection numbers by distinguishing contributions from surfaces with different numbers of boundary components, and we calculate all these numbers. We then construct a matrix model for the generating series of the refined open intersection numbers and conjecture that it is equivalent to the Kontsevich-Penner matrix model. An evidence for the conjecture is presented. Another refinement of the open intersection numbers, which describes the distribution of the boundary marked points on the boundary components, is also discussed.

On the complete intersection conjecture of Murthy

Suppose A=k[X1,X2,…,Xn] is a polynomial ring over a field k and I is an ideal in A. M.P. Murthy conjectured that μ(I)=μ(I/I2), where μ denotes the minimal number of generators. Recently, Fasel [3] settled this conjecture, affirmatively, when k is an infinite perfect field, with 1/2∈k (always). We are able to do the same, when k is an infinite field. In fact, we prove similar results for ideals I in a polynomial ring A=R[X], that contains a monic polynomial and R is essentially smooth algebra over an infinite field k, or R is a regular ring over a perfect field k.

A Hilton–Milner-type theorem and an intersection conjecture for signed sets

A family A of sets is said to be intersecting if any two sets in A intersect (i.e. have at least one common element). A is said to be centred if there is an element common to all the sets in A; otherwise, A is said to be non-centred. For any r∈[n]:={1,…,n} and any integer k≥2, let Sn,r,k be the family {{(x1,y1),…,(xr,yr)}:x1,…,xr are distinct elements of [n], y1,…,yr∈[k]} of k-signedr-sets on[n]. Let m:=max{0,2r−n}. We establish the following Hilton–Milner-type theorems, the second of which is proved using the first:(i) If A1 and A2 are non-empty cross-intersecting (i.e. any set in A1 intersects any set in A2) sub-families of Sn,r,k, then |A1|+|A2|≤nrkr−∑i=mrri(k−1)in−rr−ikr−i+1. (ii) If A is a non-centred intersecting sub-family of Sn,r,k, 2≤r≤n, then |A|≤{n−1r−1kr−1−∑i=mr−1ri(k−1)in−1−rr−1−ikr−1−i+1if r<n;kr−1−(k−1)r−1+k−1if r=n.We also determine the extremal structures. (ii) is a stability theorem that extends Erdős–Ko–Rado-type results proved by various authors. We then show that (ii) leads to further evidence for an intersection conjecture suggested by the author about general signed set systems.

Three-matching intersection conjecture for perfect matching polytopes of small dimensions

In the study of Fulkerson’s conjecture, Fan and Raspaud (1994) proposed a weaker conjecture: in every bridgeless cubic graph G, there are three perfect matchings M1, M2, and M3 such that M1∩M2∩M3=0̸. In this note we prove that this conjecture is true if the dimension of the perfect matching polytope of G is at most 9. This includes the class of bridgeless cubic graphs with the property that all the vertices of the corresponding perfect matching polytope are affinely independent.

Almost regular sequences and the monomial conjecture

This conjecture has assumed a central role since it has a very simple statement and it implies several other important conjectures, notably the Canonical Element Conjecture, for rings of positive or mixed characteristic. In fact, when this conjecture was first announced, it had numerous further consequences, some of which, such as the New Intersection Conjecture, were proved later by different means. We refer to Hochster [6] and [10] for descriptions of these conjectures and their status at various times. The Monomial Conjecture is almost trivial for rings that contain the rational numbers and is not difficult for rings of positive charateristic, but it is still an open problem for rings of mixed characteristic. The most recent advance was made by Ray Heitmann [5], who proved it in mixed characteristic in dimension 3.

Intersecting Designs

We prove the intersection conjecture for designs: For any complete graph Kr there is a finite set of positive integers M(r) such that for every n>n0(r), if Kn has a Kr-decomposition (namely a 2-(n, r, 1) design exists) then there are two Kr-decompositions of Kn having exactly q copies of Kr in common for every q belonging to the set[formula]. In fact, this result is a special case of a much more general result, which determines the existence of k distinct Kr-decompositions of Kn which have q elements in common, and all other elements of any two of the decompositions share at most one edge in common.

A Note on the First Non-Vanishing Local Cohomology Module

A case of the Weak Intersection Conjecture and the minimality of Koszul relations in a certain resolution are proved.

Canonical elements in local cohomology modules and the direct summand conjecture

One of the objectives of this paper is to show that the usual homological consequences of the existence of big Cohen-Macaulay (henceforth, C-M) modules (e.g., the new intersection conjecture of Peskine-Szpiro and Roberts and the Evans-Griffith syzygy conjecture) follow from the direct summand conjecture when the residual characteristic of the local ring is positive. This gives a new and substantially more elementary proof of the standard homological conjectures in case the characteristic of the ring itself is positive, and reduces the general case of all these conjectures to one rather down-to-earth conjecture. Of course, this places the direct summand conjecture in a position of central importance, so that it now merits an allout attack. Some partial results on this problem are given in Section 6. (The conjecture asserts that a regular Noetherian ring R is a direct summand (as an R-module) of every module-finite extension ring S 3 R.) The other main objective of this paper is to formulate and develop a theory of certain “canonical elements” in the local cohomology of special modules of syzygies. (Neither the modules of syzygies nor the induced maps between them are canonical, but the identification between the canonical elements is independent of the choices.) In particular, a canonical element qR E HZ (syz” K) is associated (see Section 3 for details) with each ndimensional local ring (R, m, K) (rings are commutative, associative, with identity; “local ring” means Noetherian ring with a unique maximal ideal). The conjecture that qR is nonzero for all local rings R turns out to be equivalent to the direct summand conjecture: for a given R, an infinite family of cases of the latter implies the former.

Cohen-Macaulay complexes and an analytic proof of the new intersection conjecture

A definition of a Cohen-Macaulay complex is given so that the existence of such a complex implies the New Intersection Conjecture. It is then shown that the Grauert-Riemenschneider vanishing theorem implies that Cohen-Macaulay complexes exist for local rings of points of complex algebraic varieties.