Monomial Maps Research Articles

Degree growth of monomial maps and McMullen's polytope algebra

We compute all dynamical degrees of monomial maps by interpreting them as mixed volumes of polytopes. By exploiting further the isomorphism between the polytope algebra of P. McMullen and the universal cohomology of complete toric varieties, we construct invariant positive cohomology classes when the dynamical degrees have no resonance.

Stabilization of monomial maps

A monomial (i.e. equivariant) selfmap of a toric variety is called stable if its action on the Picard group commutes with iteration. Generalizing work of Favre to higher dimensions, we show that under suitable conditions, a monomial map can be made stable by refining the underlying fan. In general, the resulting toric variety has quotient singularities; in dimension two we give criteria for when it can be chosen smooth, as well as examples when it cannot.

Normal forms, hermitian operators, and CR maps of spheres and hyperquadrics

We prove and organize some results on the normal forms of Hermitian operators composed with the Veronese map. We apply this general framework to prove two specific theorems in CR geometry. First, extending a theorem of Faran, we classify all real-analytic CR maps between any hyperquadric in $\C^2$ and any hyperquadric in $\C^3$, resulting in a finite list of equivalence classes. Second, we prove that all degree-two CR maps of spheres in all dimensions are spherically equivalent to a monomial map, thus obtaining an elegant classification of all degree-two CR sphere maps.

Combinatorics of Cremona monomial maps

One studies Cremona monomial maps by combinatorial means. Among the results is a simple integer matrix theoretic proof that the inverse of a Cremona monomial map is also defined by monomials of fixed degree, and moreover, the set of monomials defining the inverse can be obtained explicitly in terms of the initial data. A neat consequence is drawn for the plane Cremona monomial group, in particular the known result saying that a plane Cremona (monomial) map and its inverse have the same degree. Included is a discussion about the computational side and/or implementation of the combinatorial invariants stemming from these questions.

Monomial maps on [formula omitted] and their arithmetic dynamics

We say that a rational map on P n is a monomial map if it can be expressed in some coordinate system as [ F 0 : ⋯ : F n ] where each F i is a monomial. We consider arithmetic dynamics of monomial maps on P 2 . In particular, as Silverman (1993) explored for rational maps on P 1 , we determine when orbits contain only finitely many integral points. Our first result is that if some iterate of a monomial map on P 2 is a polynomial, then the first such iterate is 1, 2, 3, 4, 6, 8, or 12. We then completely determine all monomial maps whose orbits always contain just finitely many integral points. Our condition is based on the exponents in the monomials. In cases when there are finitely many integral points in all orbits, we also show that the sizes of the numerators and the denominators are comparable. The main ingredients of the proofs are linear algebra, such as Perron–Frobenius theorem.

Cremona maps defined by monomials

Cremona maps defined by monomials of degree 2 are thoroughly analyzed and classified via integer arithmetic and graph combinatorics. In particular, the structure of the inverse map to such a monomial Cremona map is made very explicit as is the degree of its monomial defining coordinates. As a special case, one proves that any monomial Cremona map of degree 2 has inverse of degree 2 if and only if it is an involution up to permutation in the source and in the target. This statement is subsumed in a recent result of L. Pirio and F. Russo, but the proof is entirely different and holds in all characteristics. One unveils a close relationship binding together the normality of a monomial ideal, monomial Cremona maps and Hilbert bases of polyhedral cones.

Algebraic stability and degree growth of monomial maps

Given a rational monomial map, we consider the question of finding a toric variety on which it is algebraically stable. We give conditions for when such variety does or does not exist. We also obtain several precise estimates of the degree sequences of monomial maps.

Polynomials Constant on a Hyperplane and CR Maps of Hyperquadrics

We prove a sharp degree bound for polynomials constant on a hyperplane with a fixed number of distinct monomials for dimensions 2 and 3. We study the connection with monomial CR maps of hyperquadrics and prove similar bounds in this setup with emphasis on the case of spheres. The results support generalizing a conjecture on the degree bounds to the more general case of hyperquadrics.

Computing the support of monomial iterated mapping cones

In this paper we compute and manipulate the support of monomial resolutions based on iterated mapping cones. We derive in this way algorithms to obtain homological and numerical invariants of monomial ideals without actually computing their resolution. Our computations include Betti diagrams, Hilbert series and irreducible decompositions. The algorithms derived by the method presented in the paper are efficient in practice as shown by experiments.

Linear recurrences in the degree sequences of monomial mappings

AbstractLet A be an integer matrix, and let fA be the associated monomial map. We give a connection between the eigenvalues of A and the existence of a linear recurrence relation in the sequence of degrees.

Corrigendum to ‘Degree-growth of monomial maps’ (Ergodic Theory and Dynamical Systems 27(5) (2007), 1375–1397)

Corrigendum to ‘Degree-growth of monomial maps’ (<i>Ergodic Theory and Dynamical Systems</i> 27(5) (2007), 1375–1397)

Degree-growth of monomial maps

AbstractFor projectivizations of rational maps, Bellon and Viallet defined the notion of algebraic entropy using the exponential growth rate of the degrees of iterates. We want to call this notion to the attention of dynamicists by computing algebraic entropy for certain rational maps of projective spaces (Theorem 6.2) and comparing it with topological entropy (Theorem 5.1). The particular rational maps we study are monomial maps (Definition 1.2), which are closely related to toral endomorphisms. Theorems 5.1 and 6.2 that imply that the algebraic entropy of a monomial map is always bounded above by its topological entropy, and that the inequality is strict if the defining matrix has more than one eigenvalue outside the unit circle. Also, Bellon and Viallet conjectured that the algebraic entropy of every rational map is the logarithm of an algebraic integer, and Theorem 6.2 establishes this for monomial maps. However, a simple example using a monomial map shows that a stronger conjecture of Bellon and Viallet is incorrect, in that the sequence of algebraic degrees of the iterates of a rational map of projective space need not satisfy a linear recurrence relation with constant coefficients.

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].

A sharp bound for the degree of proper monomial mappings between balls

The authors prove that a proper monomial holomorphic mapping from the two-ball to the N-ball has degree at most 2N-3, and that this result is sharp. The authors first show that certain group-invariant polynomials (related to Lucas polynomials) achieve the bound. To establish the bound the authors introduce a graph-theoretic approach that requires determining the number of sinks in a directed graph associated with the quotient polynomial. The proof also relies on a result of the first author that expresses all proper polynomial holomorphic mappings between balls in terms of tensor products.

Monomialization of strongly prepared morphisms from nonsingular n-folds to surfaces

Monomialization of morphisms is the problem of transforming a mapping into a monomial mapping by blowing up a chain of nonsingular subvarieties in its domain and image. Some articles on this and related problems are [1–7]. Consider the following basic example. Let Φ :Ank →Ak be a morphism of affine spaces over field k. Then Φ is given by a collection of polynomials f1, . . . , fm in n variables: y1 = f1(x1, . . . , xn), .. ym = fm(x1, . . . , xn).

On the monomial birational maps of the projective space

We describe the group structure of monomial Cremona transformations. It follows that every element of this group is a product of quadratic monomial transformations, and geometric descriptions in terms of fans.

ON ERGODIC BEHAVIOR OF p-ADIC DYNAMICAL SYSTEMS

Monomial mappings, x ↦ xn, are topologically transitive and ergodic with respect to Haar measure on the unit circle in the complex plane. In this paper we obtain an analogous result for monomial dynamical systems over p-adic numbers. The process is, however, not straightforward. The result will depend on the natural number n. Moreover, in the p-adic case we will not have ergodicity on the unit circle, but on the circles around the point 1.

On a Question of Igusa: Towards a Theory of Several Variable Asymptotic Expansions: I

HTML view is not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button. A question of Igusa (from 1978) inquires about the singular behavior of the singular series, determined by a polynomial mapping ${\mathbf P}$ :K$^n$→K$^m$, m[les ]n, where K is a local field of characteristic zero. This paper describes in geometric terms the singularities of the singular series for two classes of polynomial maps ${\mathbf P}$=(P$_1$, P$_2$):K$^n$→K$^2$. The main result, which makes possible this description, is a type of uniformization of ${\mathbf P}$ by finitely many monomial maps μ(${\mathbf x}$)=(${\mathbf x}$$^$$_1$, ${\mathbf x}$$^ $$_2$), such that rank ($^$$_1$$_$$_2$)= 2. This is proved using resolution of singularities. Using this result, nontrivial estimates of oscillatory integrals with phase λ$_1$P$_1$+ λ$_2$P$_2$ are possible. These will be described elsewhere.

THE MONOMIAL DIVISOR MIRROR MAP FOR LANDAU-GINZBURG ORBIFOLDS

We present the new explicit geometrical knowledge of the Landau-Ginzburg orbifolds, when a typical type of superpotential is considered. Relying on toric geometry, we show the one-to-one correspondence between some of the (a, c) states with U(1) charges (−1, 1) and the (1, 1) forms coming from blowing-up processes. Consequently, we find the monomial divisor mirror map for Landau-Ginzburg orbifolds. The possibility of the application to the models of other types is briefly discussed.

Polynomial and Rational Maps between Balls

Let B be the unit ball in the complex space C. Write Rat(B,B) for the space of proper rational holomorphic maps from B into B and Poly(B,B ) for the set of proper holomorphic polynomial maps from B into B . We say that F and G ∈ Rat(B,B) are equivalent if there are automorphisms σ ∈ Aut(B) and τ ∈ Aut(B ) such that F = τ ◦G◦σ. Proper holomorphic maps from B into B with N ≤ 2n − 2, that are sufficiently smooth up to the boundary, are equivalent to the identity map ([Fa] [Fr] [Hu]). In [HJX], it is shown that F ∈ Rat(B,B) with N ≤ 3n − 4 is equivalent to a quadratic monomial map, called the D’Angelo map. However, when the codimension is sufficiently large, there is plenty of room to construct rational holomorphic maps with certain arbitrariness by the work in Catlin-D’Angelo [CD]. Hence, it is reasonable to believe that after lifting the codimension restriction, many proper rational holomorphic maps are not equivalent to proper holomorphic polynomial maps. In the last paragraph of the paper [DA], D’Angelo gave a philosophic discussion on this matter. However, the problem of determining if an explicit proper rational holomorphic map is equivalent to a polynomial map does not seem to have been studied so far. This short paper is concerned with such a problem. We will first give a simple and explicit criterion when a rational holomorphic map is equivalent to a holomorphic polynomial map. With the help of the classification result in [CJX], this criterion is used in §3 to show that proper rational holomorphic maps from B into B of degree two are equivalent to