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Kangning Wei

Publications and source records attributed to Kangning Wei.

6 recordsLinked to original sources
arXiv2024

Discrete Painlev\'e equations and pencils of quadrics in $\mathbb P^3$

Discrete Painlev\'e equations constitute a famous class of integrable non-autonomous second order difference equations. A classification scheme proposed by Sakai interprets a discrete Painlev\'e equation as a birational map between generalized Halphen surfaces (surfaces obtained from $\mathbb P^1\times\mathbb P^1$ by blowing up at eight points). We propose a novel geometric interpretation of discrete Painlev\'e equations, where the family of generalized Halphen surfaces is replaced by a pencil of quadrics in $\mathbb P^3$. A discrete Painlev\'e equation is viewed as an autonomous birational transformation of $\mathbb P^3$ that preserves the pencil and maps each quadric of the pencil to a different one, according to a M\"obius transformation of the pencil parameter. Thus, our scheme is based on the classification of pencils of quadrics in $\mathbb P^3$.

nlin.SI↗
arXiv2023

Dynamical degrees of birational maps from indices of polynomials with respect to blow-ups II. 3D examples

The goal of this paper is the exact computation of the degrees $\text{deg}(f^n)$ of the iterates of birational maps $f: \mathbb{P}^N \dashrightarrow \mathbb{P}^N$. In the preceding companion paper, a new method has been proposed based on the use of indices of polynomials associated to the local blow-ups used to resolve contractions of hypersurfaces by $f$, and on the control of the factorization of pull-backs of polynomials. This leads to recurrence relations for the degrees and the indices. We apply this method to several illustrative examples in three dimensions. These examples demonstrate the flexibility of the method which, in particular, does not require the construction of an algebraically stable lift of $f$, unlike the previously known methods based on the Picard group.

math.DS↗
arXiv2023

A three-dimensional generalization of QRT maps

We propose a geometric construction of three-dimensional birational maps that preserve two pencils of quadrics. The maps act as compositions of involutions, which, in turn, act along the straight line generators of the quadrics of the first pencil and are defined by the intersections with quadrics of the second pencil. On each quadric of the first pencil, the maps act as two-dimensional QRT maps. While these maps are of a pretty high degree in general, we find geometric conditions which guarantee that the degree is reduced to 3. The resulting degree 3 maps are illustrated by two known and two novel Kahan-type discretizations of three-dimensional Nambu systems, including the Euler top and the Zhukovski-Volterra gyrostat with two non-vanishing components of the gyrostatic momentum.

nlin.SI↗
arXiv2023

Dynamical degrees of birational maps from indices of polynomials with respect to blow-ups I. General theory and 2D examples

In this paper we address the problem of computing $\text{deg}(f^n)$, the degrees of iterates of a birational map $f:\mathbb{P}^N\rightarrow\mathbb{P}^N$. For this goal, we develop a method based on two main ingredients: the factorization of a polynomial under pull-back of $f$, based on local indices of a polynomial associated to blow-ups used to resolve the contraction of hypersurfaces by $f$, and the propagation of these indices along orbits of $f$. For maps admitting algebraically stable modifications $f_X:X\rightarrow X$, where $X$ is a variety obtained from $\mathbb P^N$ by a finite number of blow-ups, this method leads to an algorithm producing a finite system of recurrent equations relating the degrees and indices of iterated pull-backs of linear polynomials. We illustrate the method by three representative two-dimensional examples. It is actually applicable in any dimension, and we will provide a number of three-dimensional examples as a separate companion paper.

math.DS↗
arXiv2020

Manin involutions for elliptic pencils and discrete integrable systems

We contribute to the algebraic-geometric study of discrete integrable systems generated by planar birational maps: (a) we find geometric description of Manin involutions for elliptic pencils consisting of curves of higher degree, birationally equivalent to cubic pencils (Halphen pencils of index 1), and (b) we characterize special geometry of base points ensuring that certain compositions of Manin involutions are integrable maps of low degree (quadratic Cremona maps). In particular, we identify some integrable Kahan discretizations as compositions of Manin involutions for elliptic pencils of higher degree.

nlin.SI↗