Algebraic Geometers Research Articles

On a generalization of the Deligne–Lusztig curve of Suzuki type and application to AG codes

Abstract In this article, Algebraic-Geometric (AG) codes and quantum codes associated with a family of curves that includes the famous Suzuki curve are investigated. The Weierstrass semigroup at some rational point is computed. Notably, each curve in the family turns out to be a Castle curve over some finite field and a weak Castle curve over its extensions. This is a relevant feature when codes constructed from the curve are considered.

Recent Trends in Algebraic Geometry

Algebraic geometry has grown into a broad subject, with many different streams often advancing quite independently of each other. Nonetheless, important advances have often come from visionary applications of ideas in one part of the subject to another. This workshop brought together leaders and future leaders in different areas of the subject, centered on geometric methods or geometric problems. It also brought together groups from different regions of the globe, in order to bridge communities of different sorts, and help new ideas quickly spread throughout algebraic geometry. Some of the best freshly-minted algebraic geometers were deliberately invited, so that they could meet their peers from around the world and learn about different perspectives on the subject.

Uniform matrix product states from an algebraic geometer's point of view

We apply methods from algebraic geometry to study uniform matrix product states. Our main results concern the topology of the locus of tensors expressed as uMPS, their defining equations and identifiability. By an interplay of theorems from algebra, geometry and quantum physics we answer several questions and conjectures posed by Critch, Morton and Hackbusch.

Guruswami-Sudan Decoding of Elliptic Codes Through Module Basis Reduction

This paper proposes the Guruswami-Sudan (GS) list decoding algorithm for one-point elliptic codes, in which the interpolation is realized by the module basis reduction (BR). Elliptic codes are a kind of algebraic-geometric (AG) codes with a genus of one. Over the same finite field, they have a greater codeword length than Reed-Solomon (RS) codes, capable of correcting more errors. The GS decoding consists of interpolation and root-finding, while the former that determines the interpolation polynomial <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathcal {Q}(\text {x}, \text {y}, \text {z})$ </tex-math></inline-formula> dominates the decoding complexity. By defining the Lagrange interpolation function over an elliptic function field, a basis of the interpolation module can be constructed. The desired Gröbner basis that contains <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathcal {Q}(\text {x}, \text {y}, \text {z})$ </tex-math></inline-formula> can be determined by reducing the constructed basis. This is namely the BR interpolation and it requires less finite field arithmetic operations than the conventional Kötter’s interpolation, facilitating the GS decoding. Re-encoding transform (ReT) is further introduced to facilitate the BR interpolation. This work also shows that both the BR interpolation and its ReT variant will have a lower complexity as the code rate <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${k}/{n}$ </tex-math></inline-formula> increases, where n and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${k}$ </tex-math></inline-formula> are the length and dimension of the code, respectively. Our numerical results demonstrate the complexity advantage of the BR interpolation over Kötter’s interpolation, and the performance advantage of elliptic codes over RS codes.

Fibonacci, Golden Ratio, and Vector Bundles

There is a family of vector bundles over the moduli space of stable curves that, while first appearing in theoretical physics, has been an active topic of study for algebraic geometers since the 1990s. By computing the rank of the exceptional Lie algebra g2 case of these bundles in three different ways, a family of summation formulas for Fibonacci numbers in terms of the golden ratio is derived.

Nested Lattice Coding With Algebraic Encoding and Geometric Decoding

Nested Lattice Coding (NLC) provides a generalized solution to Wyner-Ziv problems. Codebook of NLC is termed as the label-set. We find a significant difference between the label-set strictly satisfying coset's property and that predominantly used in existing NLC-related works. We name the former as Algebraic Label-Set (ALS), the latter as Geometric Label-Set (GLS), and present their enumeration strategies for practical implementation. Apart from random and coset binning, we name the enumeration of GLS as “geometric binning”. By applying ALS for encoding and GLS for decoding, we propose a refined NLC scheme called Nested Lattice Coding with Algebraic Encoding and Geometric Decoding (NLC-AC-GD). NLC-AC-GD has the same decoding reliability and higher compression rate compared with the traditional NLC scheme.

Multi-point codes from the GGS curves

This paper is concerned with the construction of algebraic-geometric (AG) codes defined from GGS curves. It is of significant use to describe bases for the Riemann-Roch spaces associated with some rational places, which enables us to study multi-point AG codes. Along this line, we characterize explicitly the Weierstrass semigroups and pure gaps by an exhaustive computation for the basis of Riemann-Roch spaces from GGS curves. In addition, we determine the floor of a certain type of divisor and investigate the properties of AG codes. Multi-point codes with excellent parameters are found, among which, a presented code with parameters $ [216,190,\geqslant 18] $ over $ \mathbb{F}_{64} $ yields a new record.

Flat Surfaces and Algebraic Curves

This workshop brought together two distinct communities: “flat” geometers, studying the moduli of flat surfaces, and Teichmüller dynamics, and algebraic geometers studying the moduli space of curves. While both communities study similar or often the same objects, very different viewpoints and toolboxes lead to different questions being addressed, and different progress being made. The workshop sought to educate each community about the techniques of the other, and to foster communication between the two groups. One particular focus was enumerative geometry.

Some recent results in the area of non-complete algebraic varieties

We will briefly describe the basic theory of non-complete algebraic varieties developed by Japanese algebraic geometers and some of the main contributions of Indian mathematicians in this area during the last ten years.

Subgroups of Cremona Groups

There have been recent breakthroughs in two related questions: the study of Cremona groups, and the problem of rationality of algebraic varieties. Our workshop brought together algebraic geometers, who discussed and tried to solve problems that are relevant to the classification of finite subgroups of Cremona groups. Priority was given to the following four related areas: automorphisms of algebraic varieties, birational geometry of Mori fibre spaces, and rationality problems.

Maximal linear spaces contained in the base loci of pencils of quadrics

The geometry of the Fano scheme of maximal linear spaces contained in the base locus of a pencil of quadrics has been studied by algebraic geometers when the base field is algebraically closed. In this paper, we work over an arbitrary base field of characteristic not equal to 2 and show how these Fano schemes are related to the Jacobians of hyperelliptic curves. In particular, if $B$ is the base locus of a generic pencil of quadrics in $\bbp^{2n+1}$, and $F$ is the Fano variety of $n - 1$ planes contained in $B$, then $F$ is a component of a disconnected commutative algebraic group $G = \picz(C) \dcup F \dcup \pico(C) \dcup F'$, where $C$ is the hyperelliptic curve defined by the discriminant form of the pencil. In the second half of this paper, we study regular pencils of quadrics, where the hyperelliptic curve defined by the discriminant is singular.

Interpolation-Based Low-Complexity Chase Decoding Algorithms for Hermitian Codes

Algebraic-geometric (AG) codes have good error-correction capability due to their generally large code word length. However, their decoding remains complex, preventing practical applications. Addressing the challenge, this paper proposes two interpolation-based low-complexity Chase (LCC) decoding algorithms for one of the most popular AG codes-Hermitian codes. By choosing $\eta $ unreliable symbols and realizing them with the two most likely decisions, $2^\eta $ decoding test-vectors can be formulated. The first LCC algorithm performs interpolation for the common elements of the test-vectors, producing an intermediate outcome that will be shared by the uncommon element interpolation. It eliminates the redundant computation for decoding each test-vector, resulting in a low-complexity. With an interpolation multiplicity of one, the decoding is further facilitated by removing the requirement of pre-calculating the Hermitian curve’s corresponding coefficients. The second LCC algorithm is an adaptive variant of the first algorithm, where the number of test-vectors is determined by the reliability of received information. When the channel condition improves, it can reduce the complexity without compromising the decoding performance. Simulation results show that the both LCC algorithms outperform a number of existing algebraic decoding algorithms for Hermitian codes. Finally, our complexity analysis will reveal the proposals’ low-complexity feature.

AG codes and AG quantum codes from the GGS curve

In this paper, algebraic-geometric (AG) codes associated with the GGS maximal curve are investigated. The Weierstrass semigroup at all \(\mathbb F_{q^2}\)-rational points of the curve is determined; the Feng-Rao designed minimum distance is computed for infinite families of such codes, as well as the automorphism group. As a result, some linear codes with better relative parameters with respect to one-point Hermitian codes are discovered. Classes of quantum and convolutional codes are provided relying on the constructed AG codes.

Number theory meets high energy physics

Feynman amplitudes in perturbative quantum field theory are being expressed in terms of an algebra of functions, extending the familiar logarithms, and associated numbers—periods. The study of these functions (including hyperlogarithms) and numbers (like the multiple zeta values), that dates back to Leibniz and Euler, has attracted anew the interest of algebraic geometers and number theorists during the last decades. The two originally independent developments are recently coming together in an unlikely collaboration between particle physics and what were regarded as the most abstruse branches of mathematics.

Complete spelling rules for the Monster tower over three-space

The Monster tower, also known as the Semple tower, is a sequence of manifolds with distributions of interest to both differential and algebraic geometers. Each manifold is a projective bundle over the previous. Moreover, each level is a fiber compactified jet bundle equipped with an action of finite jets of the diffeomorphism group. There is a correspondence between points in the tower and curves in the base manifold. These points admit a stratification which can be encoded by a word called the RVT code. Here, we derive the spelling rules for these words in the case of a three dimensional base. That is, we determine precisely which words are realized by points in the tower. To this end, we study the incidence relations between certain subtowers, called Baby Monsters, and present a general method for determining the level at which each Baby Monster is born. Here, we focus on the case where the base manifold is three dimensional, but all the methods presented generalize to bases of arbitrary dimension.

Two-Point AG Codes on the GK Maximal Curves

We determine the Weierstrass semigroup of a pair of certain rational points on the Giulietti-Korchmaros maximal curves. We use this semigroup to obtain two-point algebraic geometric (AG) codes with better parameters than comparable one-point AG codes arising from these curves. These parameters are new records in the MinT’s tables.

An Iterative Algebraic Geometric Approach for Identification of Switched ARX Models with Noise

AbstractThe algebraic geometric (AG) approach has been used to identify switched auto regressive exogenous (SARX) models in hybrid systems, and it has several advantages over other SARX identification methods. This paper is focused on improving the estimation accuracy of the AG approach for systems corrupted with indispensable noises. A stochastic hybrid decoupling polynomial (SHDP) is constructed by reformulating the hybrid decoupling polynomial (HDP) used in the original AG method. An iterative scheme is developed to estimate parameters of the SHDP, which are used to calculate the SARX model parameters. This estimation involves linear regression with multiplicative noises, therefore a novel approach is proposed to solve this regression problem. Then, the parameters are recovered from the SHDP. Finally, all these steps for SARX model identification are summarized in an algorithm called the iterative algebraic geometric (IAG) approach. Simulations and experimental validation results are shown to demonstrate the effectiveness of and the improvement made by the proposed IAG method.

Classification of multipartite systems featuring only and genuine entangled states

We present several multipartite quantum systems featuring the same type of genuine (tripartite) entanglement. Based on a geometric interpretation of the so-called and states we show that the classification of all multipartite systems featuring those and only those two classes of genuine entanglement can be deduced from earlier work of algebraic geometers. In this paper, non-entangled pure states are identified with the highest weight orbit of the SLOCC group acting on the corresponding Hilbert space.

Preface Issue 4-2015

In their survey article on “Stochastic PDEs and lack of regularity”, Dirk Blomker and Marco Romito discuss a nonlinear partial differential equation which may serve as a phenomenological model for the growth of an amorphous surface. This equation contains an additional noise term which models fluctuations of the flow of incoming particles. The authors mention that similar equations are used to model the formation of sand or snow ripples and that their model has striking similarities to the Kadar-Parisi-Zhang (KPZ) equation. While it is possible to prove local (in time) existence of strong solutions, which are also unique, the problem of whether they are global (in time) or may blow up in finite time remains an important open problem. In one space dimension (i.e., assuming invariance with respect to the second spatial variable) the authors prove the existence of global (in time) weak solutions where, however, uniqueness is left open. Similar difficulties due to the lack of regularity arise also e.g. in the time-dependent 3D-Navier-Stokes equations. Jacques Tits is one of the most influential group theorist and algebraic geometers and received, among many other awards, in 1996 the Cantor medal of the Deutsche MathematikerVereinigung and in 2008 the Abel Prize of the Norwegian Academy of Science and Letters. Two years ago, Jacques Tits’ “Collected Works” appeared which are subject of this issue’s “Classics Revisited”-contribution. Linus Kramer highlights some of their contents grouping them according to their themes. Walter D. van Suijlekom gives a summary and his opinion of Frederic Paugam’s recent book “Towards the mathematics of quantum field theory”.

Access structures of hyperelliptic secret sharing schemes

In CRYPTO 2006 Chen and Cramer proposed secret sharing schemes (SSS) from algebraic–geometric (AG) codes. The schemes are ramp schemes with gap bounded by 2g, where g is the genus of the underlying curve. Subsequently, Chen, Ling and Xing explicitly gave a complete characterization of the access structures for one special and important instance-elliptic secret sharing schemes (the ones from algebraic–geometric codes associated with elliptic curves), and additionally constructed weighted threshold secret sharing schemes from algebraic curves. In elliptic SSS case, one single point on an elliptic curve was computed to determine whether a set, with size in the gap mentioned above, is qualified. In this paper, we generalize Chen, Ling and Xing's idea and method to the case where the underlying curve is a hyperelliptic curve of arbitrary genus. By the means of Cantor's algorithm, we compute a reduced divisor to determine whether a set is qualified. Moreover, we construct a weighted hyperelliptic secret sharing schemes. Thus we reduce the gap size from 2g to g−1 in both ideal and weighted hyperelliptic SSS cases. One explicit example is provided.