Algebraic Lemma Research Articles

The Yomdin–Gromov Algebraic Lemma Revisited

In 1987, Yomdin proved a lemma on smooth parametrizations of semialgebraic sets as part of his solution of Shub’s entropy conjecture for \(C^\infty \) maps. The statement was further refined by Gromov, producing what is now known as the Yomdin–Gromov algebraic lemma. Several complete proofs based on Gromov’s sketch have appeared in the literature, but these have been considerably more complicated than Gromov’s original presentation due to some technical issues. In this note, we give a proof that closely follows Gromov’s original presentation. We prove a somewhat stronger statement, where the parametrizing maps are guaranteed to be cellular. It turns out that this additional restriction, along with some elementary lemmas on differentiable functions in o-minimal structures, allows the induction to be carried out without technical difficulties.

Complex cellular structures

We introduce the notion of a complex cell, a complexification of the cells/cylinders used in real tame geometry. For $\delta\in(0,1)$ and a complex cell $\mathcal{C}$ we define its holomorphic extension $\mathcal{C}\subset\mathcal{C}^\delta$, which is again a complex cell. The hyperbolic geometry of $\mathcal{C}$ within $\mathcal{C}^\delta$ provides the class of complex cells with a rich geometric function theory absent in the real case. We use this to prove a complex analog of the cellular decomposition theorem of real tame geometry. In the algebraic case we show that the complexity of such decompositions depends polynomially on the degrees of the equations involved. Using this theory, we refine the Yomdin-Gromov algebraic lemma on $C^r$-smooth parametrizations of semialgebraic sets: we show that the number of $C^r$ charts can be taken to be polynomial in the smoothness order $r$ and in the complexity of the set. The algebraic lemma was initially invented in the work of Yomdin and Gromov to produce estimates for the topological entropy of $C^\infty$ maps. For analytic maps our refined version, combined with work of Burguet, Liao and Yang, establishes an optimal refinement of these estimates in the form of tight bounds on the tail entropy and volume growth. This settles a conjecture of Yomdin who proved the same result in dimension two in 1991. A self-contained proof of these estimates using the refined algebraic lemma is given in an appendix by Yomdin. The algebraic lemma has more recently been used in the study of rational points on algebraic and transcendental varieties. We use the theory of complex cells in these two directions. In the algebraic context we refine a result of Heath-Brown on interpolating rational points in algebraic varieties. In the transcendental context we prove an interpolation result for (unrestricted) logarithmic images of subanalytic sets.

Rodin's formula in arbitrary codimension

We extend the Rodin's formula for $p$--modulus of the family of curves in $\mathbb{R}^n$ to arbitrary codimension. The proof relies on the formula for the $p$--modulus of family of level sets of a submersion and an algebraic lemma relating Jacobi matrices of considered maps. We state appropriate examples.

On the polynomial Wolff axioms

We confirm a conjecture of Guth concerning the maximal number of $${\delta}$$ -tubes, with $${\delta}$$ -separated directions, contained in the $${\delta}$$ -neighborhood of a real algebraic variety. Modulo a factor of $${\delta^{-{\varepsilon}}}$$ , we also prove Guth and Zahl’s generalized version for semialgebraic sets. Although the applications are to be found in harmonic analysis, the proof will employ deep results from algebraic and differential geometry, including Tarski’s projection theorem and Gromov’s algebraic lemma.

New Descriptions of the Lovász Number, and the Weak Sandwich Theorem

In 1979, L. Lovász introduced the concept of an orthonormal representation of a graph, and also a related value, now popularly known as the Lovász number of the graph. One of the remarkable properties of the Lovász number is that it lies sandwiched between the stability number of the graph and the chromatic number of the complementary graph. This fact is called the sandwich theorem. In this paper, using new descriptions of the Lovász number and linear algebraic lemmas we give three proofs for a weaker version of the sandwich theorem.

Sharp Estimates for the Size of Balls in the Complement of a Hypersurface

In this paper, we make estimates for the radius of balls contained in some component of the complementary of a complete hypersurface into a space form, generalizing and improving analogous radius estimates for embedded compact hypersurfaces obtained by Blaschke, Koutroufiotis and the authors. The results are obtained using an algebraic lemma and a tangency principle related with the length of the second fundamental form. The algebraic lemma also is used to improve a result for graphs due to Hasanis–Vlachos.

The short-wave limit for nonlinear, symmetric, hyperbolic systems

For general semilinear or quasilinear symmetric, hyperbolic systems, the short-wave approximation is studied by constructing approximate solutions to the initial-value problem associated with the hyperbolic operator by means of a multiscale WKB expansion. In the diffractive optics regime, the components of the leading-order terms of the approximate solutions are shown to satisfy differential equations of KP type. Whether these equations are scalar or not depends on the polarization of the initial datum and on the asymptotic behavior of the branches of the characteristic variety of the hyperbolic operator. The asymptotic stability of the approximate solution is proved. Following J.-L. Joly, G. Métivier, and J. Rauch, the cornerstone of this study is the characteristic variety of the hyperbolic operator. It is examined here in terms of perturbation theory, which yields new proofs of the so-called algebraic lemmas of geometric optics.

Higher initial ideals of homogeneous ideals

Introduction Borel-fixed ideals Monomial orders Some algebraic lemmas Defining the higher initial ideals Representing the higher initial ideals Group action on $R^{s+1}(I)$ Describing the action on $R^{s+1}(I)$ Borel-fixedness Higher initial ideals of hyperplane sections Representing the higher initial ideals of general hyperplane sections Higher initial ideals as combinatorial structures Reading cohomological information Examples: Points and curves in $\mathbf P^3$ References.

The Gillis–Domb–Fisher correlated random walk

Correlated random walk models figure prominently in many scientific disciplines. Of fundamental importance in such applications is the development of the characteristic function of then-step probability distribution since it contains complete information on the probability structure of the process. Using a simple algebraic lemma we derive then-step characteristic function of the Gillis correlated random walk together with other related results. In particular, we present a new and simple proof of Gillis's conjecture, consider the generalization to the Gillis–Domb–Fisher walk, and examine the effect of including an arbitrary initial distribution.

The Gillis–Domb–Fisher correlated random walk

Correlated random walk models figure prominently in many scientific disciplines. Of fundamental importance in such applications is the development of the characteristic function of the n-step probability distribution since it contains complete information on the probability structure of the process. Using a simple algebraic lemma we derive the n-step characteristic function of the Gillis correlated random walk together with other related results. In particular, we present a new and simple proof of Gillis's conjecture, consider the generalization to the Gillis–Domb–Fisher walk, and examine the effect of including an arbitrary initial distribution.

On the Formal Groups Approach to Cobordism

We give a direct proof of the fact that the complex bordism ring ${U_ * }$ is a polynomial ring, based on Quillen’s approach. The proof is self-contained, apart from the assumption that the groups ${U_n}$ are all finitely generated and an algebraic lemma on formal groups.

On the formal groups approach to cobordism

We give a direct proof of the fact that the complex bordism ring U ∗ {U_ * } is a polynomial ring, based on Quillen’s approach. The proof is self-contained, apart from the assumption that the groups U n {U_n} are all finitely generated and an algebraic lemma on formal groups.

Calculations of systems unreliability by algebraic manipulation of failure event

Computations of reliability of a given system is an important field of research in reliability engineering. There are many techniques available in the literature. Some of these are based on Boolean algebra's relations. In these papers unions of events have been reduced to give the reliability of a system. Reduction of union of events involve a lot of work and give rise to a large number of terms. In this paper we have adopted a different approach. We first set up some simple algebraic lemmas on intersections and give their proofs. Next we represent the system failure event as the intersection of paths failure events. By applying our lemmas on this intersection of events step by step we are able to expand the system failure event in the form of unions of mutually exclusive events. Simple substitution of probabilities of success or failure, as the case may be, of the events involved in the union of expanded form gives the unreliability of the system. The algorithm is explained by an example.

A loop theorem for duality spaces and fibred ribbon knots

In this paper we prove a version of the loop theorem for surfaces in the boundary of a 3-dimensional duality space, i.e. a space which resembles a 3manifold only in that it satisfies the appropriate form of Poincar6-Lefschetz duality over some field of untwisted coefficients. Our motivation comes from the fact that such spaces occur as the infinite cyclic coverings of certain 4manifolds which arise in the study of knot concordance, and as the main application of our theorem we show that if a fibred knot in the 3-sphere is a ribbon knot, then its monodromy extends over a handlebody. We approach the loop theorem via the study of planar coverings of a surface, as in the original paper of Papakyriakopoulos [11] and the subsequent work of Maskit [9]. w contains a simple geometric treatment of these matters. The main result of w is that a duality space actually satisfies duality with (twisted) coefficient module the quotient of the fundamental group ring by any power of the augmentation ideal. In w 4, the results of w167 2 and 3, together with an algebraic lemma on the intersection of the powers of the augmentation ideal of a group ring, are used to prove the loop theorem for 3-dimensional duality spaces. (For the reader's convenience a proof of the algebraic temma is included as an appendix.) w contains the application to fibred ribbon knots mentioned above. In w the result of w is used to obtain a limited amount of information on some questions about knots in the boundaries of contractible 4-manifolds. In w we apply our methods to another aspect of knot concordance, and show that for any concordance with a rationally anisotropic fibred knot (see [7]) at one end, the inclusion of the complement of the knot into the complement of the concordance induces an injection of fundamental groups. For torus knots, this question was raised by Scharlemann [14].

A Construction of Regular Polygons of pq Sides Leading to a Geometric Proof of rp - sq=l

The construction of 15, 51, and 85 sided polygons is an interesting exercise when studying ruler and compass constructions. The method is based on the famous Greatest Common Divisor theorem that if p and q are relatively prime, there exist counting numbers r, s such that rp-sq= 1. The proof of this theorem, while elementary, involves descent, and working back through a messy chain of equations. The spirit of the algebraic steps is quite different from that of geometric construction. Hence it is interesting in a geometry class to use a proof less dependent on an algebraic lemma. Indeed, a proof of the lemma falls out as a corollary.