Codimension and projective dimension up to symmetry

Symmetric ideals in increasingly larger polynomial rings that form an ascending chain are investigated. We focus on the asymptotic behavior of codimensions and projective dimensions of ideals in such a chain. If the ideals are graded it is known that the codimensions grow eventually linearly. Here this result is extended to chains of arbitrary symmetric ideals. Moreover, the slope of the linear function is explicitly determined. We conjecture that the projective dimensions also grow eventually linearly. As part of the evidence we establish two non-trivial lower linear bounds of the projective dimensions for chains of monomial ideals. As an application, this yields Cohen–Macaulayness obstructions.     

An example of a variety of linear algebras with fractional-polynomial growth

The asymptotic behavior of codimensions of identities of nonassociative algebras is studied in the paper. It is shown that in contrast with the associative case a polynomial growth of codimensions can result in a fractional exponent of a polynomial degree.     

Graded polynomial identities and codimensions: Computing the exponential growth

Let G be a finite abelian group and A a G-graded algebra over a field of characteristic zero. This paper is devoted to a quantitative study of the graded polynomial identities satisfied by A. We study the asymptotic behavior of , n=1,2,…, the sequence of graded codimensions of A and we prove that if A satisfies an ordinary polynomial identity, exists and is an integer. We give an explicit way of computing such integer by proving that it equals the dimension of a suitable finite dimension semisimple G×Z₂-graded algebra related to A.     

Identities of *-superalgebras and almost polynomial growth

We study the growth of the codimensions of a *-superalgebra over a field of characteristic zero. We classify the ideals of identities of finite dimensional algebras whose corresponding codimensions are of almost polynomial growth. It turns out that these are the ideals of identities of two algebras with distinct involutions and gradings. Along the way, we also classify the finite dimensional simple *-superalgebras over an algebraically closed field of characteristic zero.     

On minimal disjoint degenerations of modules over tame path algebras

For representations of tame quivers the degenerations are controlled by the dimensions of various homomorphism spaces. Furthermore, there is no proper degeneration to an indecomposable. Therefore, up to common direct summands, any minimal degeneration from M to N is induced by a short exact sequence 0→U→M→V→0 with indecomposable ends that add up to N. We study these ‘building blocs’ of degenerations and we prove that the codimensions are bounded by two. Therefore, a quiver is Dynkin resp. Euclidean resp. wild iff the codimension of the building blocs is one resp. bounded by two resp. unbounded. We explain also that for tame quivers the complete classification of all the building blocs is a finite problem that can be solved with the help of a computer.     

Ultrafilters and Ramsey-type shadowing phenomena in topological dynamics

The graded exponent is an important invariant of group graded PI-algebras. In this paper we study a specific elementary grading on the algebra of upper triangular matrices UT _n , compute its codimensions, and use this grading to find the asymptotic behaviour of the codimensions of any elementary grading on UT _n , for any group. Moreover, we extend this to the Lie case, and obtain, for any elementary grading on the Lie algebra UT_n^(?), an upper bound and a lower bound for the asymptotic behaviour of its codimensions. Also, we obtain the graded exponent of any grading on UT_n^(?) and for any grading on the Jordan algebra UJ _n .It turns out that the graded exponent for UT _n , considered as an associative, Jordan or Lie algebra, for any grading, coincides with the exponent of the ordinary case. In the associative case, the asymptotic behaviour of the codimensions of any grading on UT _n coincides with the asymptotic behaviour of the ordinary codimensions. But this is not the case for the graded asymptotics of the codimensions of the Lie algebra UT_n^(?).     

A Cubic Model of a Real Variety

In the paper a cubic model is constructed for a germ of a real subvariety in a complex space. It is shown that in its range of codimensions this model possesses the full spectrum of properties similar to well-known properties of tangent quadrics.     

The codimensions of aPi-algebra

We simplify the numerical calculations given in a previous paper by Regev and obtain a much better estimation for the sequence of codimensions of aPI-algebra.     

Codimensions of algebras and growth functions

Let A be an algebra over a field F of characteristic zero and let cn(A), , be its sequence of codimensions. We prove that if cn(A) is exponentially bounded, its exponential growth can be any real number >1. This is achieved by constructing, for any real number α>1, an F-algebra Aα such that exists and equals α. The methods are based on the representation theory of the symmetric group and on properties of infinite Sturmian and periodic words.     

Complete submanifolds with parallel mean curvature vector in hyperbolic spaces

In this paper we proved a better estimate as well as generalized to higher codimensions of a theorem of Y.B. Shen on complete submanifolds with parallel mean curvature vector in a hyperbolic space.     

Boundary Regularity for Minimal Graphs of Higher Codimensions

In this paper, the authors derive H¨older gradient estimates for graphic functions of minimal graphs of arbitrary codimensions over bounded open sets of Euclidean space under some suitable conditions.     

Asymptotic growth of codimensions of identities of associative algebras

Numerical characteristics of identities of associative and non-associative algebras are studied in the paper. It is announced that the sequence of codimensions of an arbitrary associative PI-algebra asymptotically increases and that this is not true in the general non-associative case.     

Bogdanov‐Takens bifurcations of codimensions 2 and 3 in a Leslie‐Gower predator‐prey model with Michaelis‐Menten–type prey harvesting

The Bogdanov‐Takens bifurcations of a Leslie‐Gower predator‐prey model with Michaelis‐Menten–type prey harvesting were studied. In the paper “Diff. Equ. Dyn. Syst. 20(2012), 339‐366,” Gupta et al proved that the Leslie‐Gower predator‐prey model with Michaelis‐Menten–type prey harvesting has rich dynamics. Some equilibria of codimension 1 and their bifurcations were discussed. In this paper, we find that the model has an equilibrium of codimensions 2 and 3. We also prove analytically that the model undergoes Bogdanov‐Takens bifurcations (cusp cases) of codimensions 2 and 3. Hence, the model can have 2 limit cycles, coexistence of a stable homoclinic loop and an unstable limit cycle, supercritical and subcritical Hopf bifurcations, and homoclinic bifurcation of codimension 1 as the values of parameters vary. Moreover, several numerical simulations are conducted to illustrate the validity of our results.     

The Bertini irreducibility theorem for higher codimensional slices

In [3], Poonen and Slavov recently developed a novel approach to Bertini irreducibility theorems over an arbitrary field, based on random hyperplane slicing. In this paper, we extend their work by proving an analogous bound for the dimension of the exceptional locus in the setting of linear subspaces of higher codimensions.     

Spherical Minimal Immersions with Prescribed Codimension

We describe a general construction of manufacturing new spherical minimal immersions between round spheres out of old ones. The new immersions have higher domain dimension and degree and the construction has a precise control on the codimension. Applied to classified and recent examples, the construction gives an abundance of new spherical minimal immersions with prescribed codimensions.Mathematics Subject Classifications (2000). 53C42     

Higher codimension cycles on the Hilbert scheme of three points on the projective plane

In this paper, we study the higher codimensional cycle structure of the Hilbert scheme of three points in the projective plane. In particular, we compute all Chern/Segre classes of all tautological bundles on it and compute the nef (effective) cones of cycles in codimensions 2 and 3 (dimensions 2 and 3).     

Free CR distributions

There are only some exceptional CR dimensions and codimensions such that the geometries enjoy a discrete classification of the pointwise types of the homogeneous models. The cases of CR dimensions n and codimensions n ² are among the very few possibilities of the so-called parabolic geometries. Indeed, the homogeneous model turns out to be PSU(n+1,n)/P with a suitable parabolic subgroup P. We study the geometric properties of such real (2n+n ²)-dimensional submanifolds in for all n > 1. In particular, we show that the fundamental invariant is of torsion type, we provide its explicit computation, and we discuss an analogy to the Fefferman construction of a circle bundle in the hypersurface type CR geometry.     

Identities on Clifford algebras

We obtain some estimates of the codimensions and PI-exponent of identities on the Clifford algebras associated to a quadratic form of a given rank, find a hook where the cocharacters of Clifford algebras lie, exhibit an identity that holds on the Clifford algebras of a given rank, and describe the multilinear identities of minimal degree for the Clifford algebras of rank 1.     

Classification of (2,2)-stable unfoldings of map germs with codimensions ≤ 4

In this paper we classify (2,2)-stable unfoldings of map germs with codimensions 4 by means of the algorithm given in [4] by Wassermann.     

The exponential growth of codimensions for Capelli identities

By applying the theorem that every positive integer is a sum of four squares, we calculate the exponential growth of the codimensions for the relatively free algebra satisfying Capelli identities. Work partially supported by RFFI grants 96-01-00146 and 98-01-01020. Work partially supported by ISF grant 6629/1. Work partially supported by RFFI grants 96-01-00146 and 96-15-96050.