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In this paper, we study the existence of multiple normalized solutions to the following Schrödinger–Poisson system with general nonlinearities: \begin{equation*} \begin{cases} -\varepsilon^2\Delta u+V(x)u+\phi u=f(u)+\lambda u & \hbox{in $\mathbb{R}^3$,} \\ -\varepsilon^2\Delta\phi=u^2& \hbox{in $\mathbb{R}^3$,}\\ \int_{\mathbb{R}^3}|u|^2{\rm d}x=\varepsilon^3 a^2,\ \end{cases} \end{equation*} where $\varepsilon$, $a>0$, $\lambda\in\mathbb{R}$ is an unknown parameter that appears as a Lagrange multiplier, $V(x):\mathbb{R}^3 \rightarrow [0,\infty)$ is a continuous function, and $f$ is a differentiable function satisfying $L^2$-subcritical growth. Through using the minimization techniques and the Lusternik–Schnirelmann category, we prove that the numbers of normalized solutions are related to the topology of the set where the potential $V(x)$ attains its minimum value.

This article is concerned with the existence of multiple normalized solutions for a class of Choquard equations with a parametric perturbation −ε2Δu+V(x)u=λu+ε−α(Iα*F(u))f(u),x∈RN,∫RN∣u∣2dx=a2εN,\left\{\begin{array}{ll}-{\varepsilon }^{2}\Delta u+V\left(x)u=\lambda u+{\varepsilon }^{-\alpha }\left({I}_{\alpha }* F\left(u))f\left(u),\hspace{1.0em}& x\in {{\mathbb{R}}}^{N},\\ \mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}{| u| }^{2}{\rm{d}}x={a}^{2}{\varepsilon }^{N},\hspace{1.0em}& \end{array}\right. where a>0a\gt 0 is a constant, ε>0\varepsilon \gt 0 is a parameter, N≥3N\ge 3, α∈(0,N)\alpha \in \left(0,N), λ∈R\lambda \in {\mathbb{R}} is unknown and appears as a Lagrange multiplier, ff is a continuous function with L2{L}^{2}-subcritical growth, and V:RN→[0,∞)V:{{\mathbb{R}}}^{N}\to \left[0,\infty ) is a continuous function, satisfying del Pino and Felmer’s local conditions. With the help of the penalization method, and Lusternik-Schnirelmann theory, we investigate the relationship between the number of positive normalized solutions and the topology of the set, where the potential VV attains its minimum value if the parameter ε>0\varepsilon \gt 0 is small.

We study the existence of minimal networks in the unit sphere S d and the unit ball B d of R d endowed with Riemannian metrics close to the standard ones. We employ a finite-dimensional reduction method, modelled on the configuration of θ -networks in S d and triods in B d , jointly with the Lusternik–Schnirelmann category. [ABSTRACT FROM AUTHOR]

In this article, we are first concerned with the existence of multiple normalized solutions to the following fractional p-Laplace problem: { (- Δ) p s ⁢ v + 풱 ⁢ (ξ ⁢ x) ⁢ | v | p - 2 ⁢ v = λ ⁢ | v | p - 2 ⁢ v + f ⁢ (v) in ⁢ ℝ N , ∫ ℝ N | v | p ⁢ 푑 x = a p , where a , ξ > 0 , p ≥ 2 , λ ∈ ℝ is an unknown parameter that appears as a Lagrange multiplier, 풱 : ℝ N → [ 0 , ∞) is a continuous function, and f is a continuous function with L p -subcritical growth. We prove that there exists the multiplicity of solutions by using the Lusternik–Schnirelmann category. Next, we show that the number of normalized solutions is at least the number of global minimum points of 풱 , as ξ is small enough via Ekeland's variational principle. [ABSTRACT FROM AUTHOR]

In this paper we prove the existence of multiple positive solutions for a quasilinear elliptic problem with unbalanced growth in expanding domains by using variational methods and the Lusternik–Schnirelmann category theory. Based on the properties of the category, we introduce suitable maps between the expanding domains and the critical levels of the energy functional related to the problem, which allow us to estimate the number of positive solutions by the shape of the domain. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence of multiple normalized solutions for a Choquard equation involving fractional p-Laplacian in R N . With the help of variational methods, minimization techniques, and the Lusternik–Schnirelmann category, the existence of multiple normalized solutions is obtained for the above problem. [ABSTRACT FROM AUTHOR]

In this study, we compute some examples of sectional category secat(f) and sectional number sec(f) for continuous maps f related to finite spaces. Moreover, we introduce an invariant secatk(f) for a map f between finite spaces using the k-th barycentric subdivision and show the equality secatk(f) = secat(B(f)) for sufficiently large k, where B(f) is the induced map on the associated polyhedra. [ABSTRACT FROM AUTHOR]

Over a local ring R, the theory of cohomological support varieties attaches to any bounded complex M of finitely generated R-modules an algebraic variety {\mathrm {V}}_R(M) that encodes homological properties of M. We give lower bounds for the dimension of {\mathrm {V}}_R(M) in terms of classical invariants of R. In particular, when R is Cohen–Macaulay and not complete intersection we find that there are always varieties that cannot be realized as the cohomological support of any complex. When M has finite projective dimension, we also give an upper bound for \dim {\mathrm {V}}_R(M) in terms of the dimension of the radical of the homotopy Lie algebra of R. This leads to an improvement of a bound due to Avramov, Buchweitz, Iyengar, and Miller on the Loewy lengths of finite free complexes, and it recovers a result of Avramov and Halperin on the homotopy Lie algebra of R. Finally, we completely classify the varieties that can occur as the cohomological support of a complex over a Golod ring. [ABSTRACT FROM AUTHOR]

We consider the Lorentz force equation d d t (m x ˙ 1 − | x ˙ | 2 / c 2 ) = q (E (t , x) + x ˙ × B (t , x)) , x ∈ R 3 , in the physically relevant case of a singular electric field E. Assuming that E and B are T -periodic in time and satisfy suitable further conditions, we prove the existence of infinitely many T -periodic solutions. The proof is based on a min-max principle of Lusternik-Schnirelmann type, in the framework of non-smooth critical point theory. Applications are given to the problem of the motion of a charged particle under the action of a Liénard-Wiechert potential and to the relativistic forced Kepler problem. [ABSTRACT FROM AUTHOR]

We study combinatorial and topological properties of the universal complexes X (F p n) and K (F p n) whose simplices are certain unimodular subsets of F p n . We calculate their f -vectors and the bigraded Betti numbers of their Tor-algebras, show that they are shellable, and find their applications in toric topology and number theory. We show that the Lusternick–Schnirelmann category of the moment angle complex of X (F p n) is n, provided p is an odd prime, and the Lusternick–Schnirelmann category of the moment angle complex of K (F p n) is [ n 2 ] . Based on the universal complexes, we introduce the Buchstaber invariant s p for a prime number p. [ABSTRACT FROM AUTHOR]

This paper concerns the existence of multiple solutions for a Schrödinger logarithmic equation of the form − 2 Δ u + V (x) u = u log u 2 , in ℝ N , u ∈ H 1 (ℝ N) , (P ) where V : ℝ N → ℝ is a continuous function that satisfies some technical conditions and is a positive parameter. We will establish the multiplicity of solution for (P ) by using the notion of Lusternik–Schnirelmann category, by introducing a new function space where the energy functional is C 1 . [ABSTRACT FROM AUTHOR]

For simplicial complexes and simplicial maps, the notion of being in the same contiguity class is defined as the discrete version of homotopy. In this paper, we study the contiguity distance, SD, between two simplicial maps adapted from the homotopic distance. In particular, we show that simplicial versions of LS -category and topological complexity are particular cases of this more general notion. Moreover, we present the behaviour of SD under the barycentric subdivision, and its relation with strong collapsibility of a simplicial complex. [ABSTRACT FROM AUTHOR]

In this paper we study the existence of multiple normalized solutions to the class of elliptic problems... in RN, where a, ϵ>0, λ∈R is an unknown parameter that appears as a Lagrange multiplier, V: RN→[0,∞) is a continuous function, and f is a differentiable function with L²-subcritical growth. It is proved that the numbers of normalized solutions are related to the topology of the set where the potential V attains its minimum value. In the proof our main result, we apply minimization techniques and the Lusternik-Schnirelmann category. [ABSTRACT FROM AUTHOR]

In this paper, using the notion of subdivision, the authors generalize the definition of cofibration in digital topology and show that this kind of cofibration is injective in the sense of subdivision. Meanwhile, they give the necessary condition under which a digital map is a cofibration. Furthermore, they consider the Lusternik-Schnirelmann category of digital maps in the sense of subdivision and give several fundamental homotopy properties about it. [ABSTRACT FROM AUTHOR]

We consider a geodesic problem in a manifold endowed with a Randers–Kropina metric. This is a type of a singular Finsler metric arising both in the description of the lightlike vectors of a spacetime endowed with a causal Killing vector field and in the Zermelo’s navigation problem with a wind represented by a vector field having norm not greater than one. By using Lusternik–Schnirelman theory, we prove existence of infinitely many geodesics between two given points when the manifold is not contractible. Due to the type of non-holonomic constraints that the velocity vectors must satisfy, this is achieved thanks to some recent results about the homotopy type of the set of solutions of an affine control system associated with a totally non-integrable distribution. [ABSTRACT FROM AUTHOR]

In this paper we show how the notion of the Lusternik–Schnirelmann relative category can be used to study a multiplicity problem for brake orbits in a potential well which is homeomorphic to the N-dimensional unit disk. The estimate of the relative category of the set of chords with endpoints on the (N − 1)-unit sphere was shown to the third author by Fadell and Husseini while he was visiting the University of Wisconsin at Madison. [ABSTRACT FROM AUTHOR]

We are trying to look over the Lusternik–Schnirelmann theory (L-S theory, for short) and the Topological Complexity (TC, for short) as a natural extension of the L-S theory. In particular, we focus on the impact of the ideas originated from E. Fadell and S. Husseini on both theories. More precisely, we see how their ideas on a category weight and a relative category drive the L-S theory and the TC. [ABSTRACT FROM AUTHOR]

We present a comparative study of certain invariants defined for group actions and their analogues defined for orbifolds. In particular, we prove that Fadell’s equivariant category for G-spaces coincides with the Lusternik–Schnirelmann category for orbifolds when the group is finite. [ABSTRACT FROM AUTHOR]

In this paper, we study the relation between the Lusternik– Schnirelmann category and the topological complexity of two closed oriented manifolds connected by a degree one map. [ABSTRACT FROM AUTHOR]

We define and develop a homotopy invariant notion for the sequential topological complexity of a map f : X → Y , denoted T C r (f) , that interacts with T C r (X) and T C r (Y) in the same way Jamie Scott's topological complexity map T C (f) interacts with T C (X) and T C (Y). Furthermore, we apply T C r (f) to studying group homomorphisms ϕ : Γ → Λ. In addition, we give the characterization of cohomological dimension of group homomorphisms. [ABSTRACT FROM AUTHOR]

In a recent publication [D. Govc, W. A. Marzantowicz and P. Pavešić, Estimates of covering type and the number of vertices of minimal triangulations, Discrete Comput. Geom. 63 2020, 1, 31–48], we have introduced a new method, based on the Lusternik–Schnirelmann category and the cohomology ring of a space X, that yields lower bounds for the size of a triangulation of X. In this current paper, we present an important extension that takes into account the fundamental group of X. In fact, if π 1 ⁢ (X) contains elements of finite order, then one can often find cohomology classes of high 'category weight', which in turn allow for much stronger estimates of the size of triangulations of X. We develop several weighted estimates and then apply our method to compute explicit lower bounds for the size of triangulations of orbit spaces of cyclic group actions on a variety of spaces including products of spheres, Stiefel manifolds, Lie groups and highly-connected manifolds. [ABSTRACT FROM AUTHOR]

For a graph $\Gamma$ , let $K(H_{\Gamma },\,1)$ denote the Eilenberg–Mac Lane space associated with the right-angled Artin (RAA) group $H_{\Gamma }$ defined by $\Gamma$. We use the relationship between the combinatorics of $\Gamma$ and the topological complexity of $K(H_{\Gamma },\,1)$ to explain, and generalize to the higher TC realm, Dranishnikov's observation that the topological complexity of a covering space can be larger than that of the base space. In the process, for any positive integer $n$ , we construct a graph $\mathcal {O}_n$ whose TC-generating function has polynomial numerator of degree $n$. Additionally, motivated by the fact that $K(H_{\Gamma },\,1)$ can be realized as a polyhedral product, we study the LS category and topological complexity of more general polyhedral product spaces. In particular, we use the concept of a strong axial map in order to give an estimate, sharp in a number of cases, of the topological complexity of a polyhedral product whose factors are real projective spaces. Our estimate exhibits a mixed cat-TC phenomenon not present in the case of RAA groups. [ABSTRACT FROM AUTHOR]

Let Gk;n be the Grassmann manifold of k-planes in Rn+k. The Lusternik-Schnirelmann category and topological complexity are important invariants of topological spaces. In this note we calculate the Lusternik-Schnirelmann category and topological complexity of certain products of Grassmannian manifolds by using cup and zero-cup length. Also we will find the lower and upper bounds of the topological complexity of some Grassmannian manifolds by the same method. [ABSTRACT FROM AUTHOR]

Digital topology is part of the ongoing endeavor to understand and analyze digitized images. With a view to supporting this endeavor, many notions from algebraic topology have been introduced into the setting of digital topology. But some of the most basic notions from homotopy theory remain largely absent from the digital topology literature. We embark on a development of homotopy theory in digital topology, and define such fundamental notions as function spaces, path spaces, and cofibrations in this setting. We establish digital analogues of basic homotopy-theoretic properties such as the homotopy extension property for cofibrations, and the homotopy lifting property for certain evaluation maps that correspond to path fibrations in the topological setting. We indicate that some depth may be achieved by using these homotopy-theoretic notions to give a preliminary treatment of Lusternik–Schnirelmann category in the digital topology setting. This topic provides a connection between digital topology and critical points of functions on manifolds, as well as other topics from topological dynamics. [ABSTRACT FROM AUTHOR]

In this paper we investigate the spectral problem in Finsler geometry. Due to the nonlinearity of the Finsler–Laplacian operator, we introduce faithful dimension pairs by means of which the spectrum of a compact reversible Finsler metric measure manifold is defined. Various upper and lower bounds of such eigenvalues are provided in the spirit of Cheng, Buser and Gromov, which extend in several aspects the results of Hassannezhad, Kokarev and Polterovich. Moreover, we construct several faithful dimension pairs based on Lusternik–Schnirelmann category, Krasnoselskii genus and essential dimension, respectively; however, we also show that the Lebesgue covering dimension pair is not faithful. As an application, we show that the Bakry–Émery spectrum of a closed weighted Riemannian manifold can be characterized by the faithful Lusternik–Schnirelmann dimension pair. [ABSTRACT FROM AUTHOR]

We use some detailed knowledge of the cohomology ring of real Grassmann manifolds Gk(ℝn) to compute zero-divisor cup-length and estimate topological complexity of motion planning for k-linear subspaces in ℝn. In addition, we obtain results about monotonicity of Lusternik–Schnirelmann category and topological complexity of Gk(ℝn) as a function of n. [ABSTRACT FROM AUTHOR]

We find conditions which ensure that the topological complexity of a closed manifold M with abelian fundamental group is nonmaximal, and see through examples that our conditions are sharp. This generalizes results of Costa and Farber on the topological complexity of spaces with small fundamental group. Relaxing the commutativity condition on the fundamental group, we also generalize results of Dranishnikov on the Lusternik–Schnirelmann category of the cofibre of the diagonal map Δ : M → M × M {\Delta:M\to M\times M} for nonorientable surfaces by establishing the nonmaximality of this invariant for a large class of manifolds. [ABSTRACT FROM AUTHOR]

Using the combinatorics of the underlying simplicial complex K, we give various upper and lower bounds for the Lusternik–Schnirelmann (LS) category of moment-angle complexes 𝒵K. We describe families of simplicial complexes and combinatorial operations which allow for a systematic description of the LS-category. In particular, we characterize the LS-category of moment-angle complexes 𝒵K over triangulated d-manifolds K for d ≤ 2, as well as higher-dimensional spheres built up via connected sum, join, and vertex doubling operations. We show that the LS-category closely relates to vanishing of Massey products in H*(𝒵K), and through this connection we describe first structural properties of Massey products in moment-angle manifolds. Some of the further applications include calculations of the LS-category and the description of conditions for vanishing of Massey products for moment-angle manifolds over fullerenes, Pogorelov polytopes and k-neighborly complexes, which double as important examples of hyperbolic manifolds. [ABSTRACT FROM AUTHOR]

Abstract We prove the existence of multiple positive solutions for a fractional Laplace problem with critical growth and sign-changing weight in non-contractible domains.

The concept of the homotopic distance and its higher analogs are introduced in [7]. In this paper we introduce some important properties of higher homotopic distances, investigate the conditions under which cat, secat and higher dimensional topological complexity categories are equal to the higher homotopic distance, and give alternative proofs, using higher homotopic distances, to some TCn-related theorems. [ABSTRACT FROM AUTHOR]

In this paper, we are interested in the following coupled critical Hartree system −Δu+λV1(x)u=α1u+βv+|x|−μ*|v|2μ*|u|2μ*−2uinℝN,−Δv+λV2(x)v=βu+α2v+|x|−μ*|u|2μ*|v|2μ*−2vinℝN,where λ > 0, β ≥ 0, α1,α2∈ℝ, 0 < μ < N, N ≥ 4, 2μ*=(2N−μ)/(N−2) are the upper critical exponents due to the Hardy‐Littlewood‐Sobolev inequality and the nonnegative potential function V1,V2∈C(ℝN,ℝ) has potential as well. By using variational methods, we prove the existence of ground state solutions and also characterize the asymptotic behavior of the solutions as the parameter λ goes to infinity. Furthermore, we are able to find the existence of multiple semitrivial weak solutions by the Lusternik‐Schnirelmann category theory. [ABSTRACT FROM AUTHOR]

We introduce a notion of categorical homotopic distance between functors by adapting the notion of homotopic distance in topological spaces, recently defined by the authors, to the context of small categories. Moreover, this notion generalizes the work on categorical LS-category of small categories by Tanaka. [ABSTRACT FROM AUTHOR]

The Schwarz genus g(ξ) of a fibration ξ : E → B is defined as the minimal integer n such that there exists a cover of B by n open sets that admit partial sections to ξ. Many important concepts, including the Lusternik-Schnirelmann category, Farber's topological complexity, and Smale-Vassiliev's complexity of algorithms can be naturally expressed as Schwarz genera of suitably chosen fibrations. In this paper we study Schwarz genus in relation with certain types of morphisms between fibrations. Our main result is the following: if there exists a fibrewise map ƒ : E → E' between fibrations ξ : E → B and ξ : E' → B which induces an n-equivalence between respective fibres for a sufficiently big n, then g(ξ) = g(ξ'). From this we derive several interesting results relating the topological complexity of a space with the topological complexities of its skeleta and subspaces (and similarly for the category). For example, we show that if a CW-complex has high topological complexity (with respect to its dimension and connectivity), then the topological complexity of its skeleta is an increasing function of the dimension. [ABSTRACT FROM AUTHOR]

We use the homotopy invariance of equivariant principal bundles to prove that the equivariant A-category of Clapp and Puppe is invariant under Morita equivalence. As a corollary, we obtain that both the equivariant Lusternik-Schnirelmann category of a group action and the invariant topological complexity are invariant under Morita equivalence. This allows a definition of topological complexity for orbifolds. [ABSTRACT FROM AUTHOR]

In this paper we study quasilinear elliptic equations defined on symmetrical expanding domains driven by the (p , q) -Laplacian and with a superlinear right-hand side. Based on the Lusternik-Schnirelmann category we prove the existence of at least γ (Ω λ ∖ { 0 }) pairs (± u) of odd weak solutions with precisely two nodal domains, where γ stands for the genus. [ABSTRACT FROM AUTHOR]

Let Crit M denote the minimal number of critical points (not necessarily non-degenerate) on a closed smooth manifold M. We are interested in the evaluation of Crit. It is worth noting that we do not know yet whether Crit M is a homotopy invariant of M. This makes the research of Crit a challenging problem. In particular, we pose the following question: given a map f : M → N of degree 1 of closed manifolds, is it true that Crit M ≥ Crit N ? We prove that this holds in dimension 3 or less. Some high dimension examples are considered. Note also that an affirmative answer to the question implies the homotopy invariance of Crit; this simple observation is a good motivation for the research. [ABSTRACT FROM AUTHOR]

We prove the equality cat (ϕ) = cd (ϕ) for epimorphisms ϕ : Γ → Λ between torsion-free, finitely generated almost nilpotent groups Γ and Λ. In addition, we prove the equality cat (ϕ) = cd (ϕ) for homomorphisms ϕ : Γ → Λ between torsion-free, finitely generated virtually nilpotent groups. [ABSTRACT FROM AUTHOR]

The covering type of a space X is a numerical homotopy invariant which in some sense measures the homotopical size of X. It was first introduced by Karoubi and Weibel (in Enseign Math 62(3–4):457–474, 2016) as the minimal cardinality of a good cover of a space Y taken among all spaces that are homotopy equivalent to X. We give several estimates of the covering type in terms of other homotopy invariants of X, most notably the ranks of the homology groups of X, the multiplicative structure of the cohomology ring of X and the Lusternik–Schnirelmann category of X. In addition, we relate the covering type of a triangulable space to the number of vertices in its minimal triangulations. In this way we derive within a unified framework several estimates of vertex-minimal triangulations which are either new or extensions of results that have been previously obtained by ad hoc combinatorial arguments. [ABSTRACT FROM AUTHOR]

We prove that the (reduced) rational sectional category of the universal fibration with fibre X, for X any space that satisfies a well-known conjecture of Halperin, equals one. [ABSTRACT FROM AUTHOR]

The Lusternik–Schnirelmann category cat and topological complexity TC are related homotopy invariants. The topological complexity TC has applications to the robot motion planning problem. We calculate the Lusternik–Schnirelmann category and topological complexity of the ordered configuration space of two distinct points in the product $G\times \mathbb{R}^{n}$ and apply the results to the planar and spatial motion of two rigid bodies in $\mathbb{R}^{2}$ and $\mathbb{R}^{3}$ respectively. [ABSTRACT FROM AUTHOR]

Let Sp (n) be the symplectic group of quaternionic (n × n) -matrices. For any 1 ≤ k ≤ n , an element A of Sp (n) can be decomposed in A = [ α T β P ] with P a (k × k) -matrix. In this work, starting from a singular value decomposition of P , we obtain what we call a relative singular value decomposition of A. This feature is well adapted for the study of the quaternionic Stiefel manifold X n , k , and we apply it to the determination of the Lusternik-Schnirelmann category of Sp (k) in X 2 k − j , k , for j = 0 , 1 , 2. [ABSTRACT FROM AUTHOR]

We study lower bounds for the number of vertices in a PL-triangulation of a given manifold M. While most of the previous estimates are based on the dimension and the connectivity of M , we show that further information can be extracted by studying the structure of the fundamental group of M and applying techniques from the Lusternik-Schnirelmann category theory. In particular, we prove that every PL-triangulation of a d -dimensional manifold (d ⩾ 3) whose fundamental group is not free has at least 3 d + 1 vertices. As a corollary, every d -dimensional homology sphere that admits a combinatorial triangulation with less than 3 d vertices is PL-homeomorphic to S d . Another important consequence is that every triangulation with small links of M is combinatorial. [ABSTRACT FROM AUTHOR]

We show a method to determine topological complexity from the fibrewise view point, which provides an alternative proof for tc (K) = 4 , where K denotes Klein bottle. [ABSTRACT FROM AUTHOR]

Abstract In this article we study the higher topological complexity TC r (X) in the case when X is an aspherical space, X = K (π , 1) and r ≥ 2. We give a characterisation of TC r (K (π , 1)) in terms of classifying spaces for equivariant Bredon cohomology. Our recent paper [7] , joint with M. Grant and G. Lupton, treats the special case r = 2. We also obtain in this paper useful lower bounds for TC r (π) in terms of cohomological dimension of subgroups of π × π × ... × π (r times) with certain properties. As an illustration of the main technique we find the higher topological complexity of Higman's group. We also apply our method to obtain a lower bound for the higher topological complexity of the right angled Artin (RAA) groups, which, as was established in [16] by a different method (in a more general situation), coincides with the precise value. We finish the paper by a discussion of the TC -generating function ∑ r = 1 ∞ TC r + 1 (X) x r encoding the values of the higher topological complexity TC r (X) for all values of r. We show that in many examples (including the case when X = K (H , 1) with H being a RAA group) the TC -generating function is a rational function of the form P (x) (1 − x) 2 where P (x) is an integer polynomial with P (1) = cat (X). [ABSTRACT FROM AUTHOR]