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The v-number of a graded ideal is an algebraic invariant introduced by Cooper et al., and originally motivated by problems in algebraic coding theory. In this paper we study the case of binomial edge ideals and we establish a significant connection between their v-numbers and the concept of connected domination in graphs. More specifically, we prove that the localization of the v-number at one of the minimal primes of the binomial edge ideal J G of a graph G coincides with the connected domination number of the defining graph, providing a first algebraic description of the connected domination number. As an immediate corollary, we obtain a sharp combinatorial upper bound for the v-number of binomial edge ideals of graphs. Lastly, building on some known results on edge ideals, we analyse how the v-number of J G behaves under Gröbner degeneration when G is a closed graph.

Mixtures of r independent distributions for two discrete random variables can be represented by matrices of nonnegative rank r. Likelihood inference for the model of such joint distributions leads to problems in real algebraic geometry that are addressed here for the first time. We characterize the set of fixed points of the Expectation-Maximization algorithm, and we study the boundary of the space of matrices with nonnegative rank at most 3. Both of these sets correspond to algebraic varieties with many irreducible components.

The 3-dimensional (3D) structure of the genome is of significant importance for many cellular processes. In this paper, we study the problem of reconstructing the 3D structure of chromosomes from Hi-C data of diploid organisms, which poses additional challenges compared to the better-studied haploid setting. With the help of techniques from algebraic geometry, we prove that a small amount of phased data is sufficient to ensure finite identifiability, both for noiseless and noisy data. In the light of these results, we propose a new 3D reconstruction method based on semidefinite programming, paired with numerical algebraic geometry and local optimization. The performance of this method is tested on several simulated datasets under different noise levels and with different amounts of phased data. We also apply it to a real dataset from mouse X chromosomes, and we are then able to recover previously known structural features.

We study solutions of difference equations in the rings of sequences and, more generally, solutions of equations with a monoid action in the ring of sequences indexed by the monoid. This framework includes, for example, difference equations on grids (for example, standard difference schemes) and difference equations in functions on words. On the universality side, we prove a version of strong Nullstellensatz for such difference equations under the assumption that the cardinality of the ground field is greater than the cardinality of the monoid and construct an example showing that this assumption cannot be omitted. On the undecidability side, we show that the following problems are undecidable:

In this paper, we construct the first provably-secure isogeny-based (partially) blind signature scheme. While at a high level the scheme resembles the Schnorr blind signature, our work does not directly follow from that construction, since isogenies do not offer as rich an algebraic structure. Specifically, our protocol does not fit into the linear identification protocol ion introduced by Hauck, Kiltz, and Loss (EUROCYRPT’19), which was used to generically construct Schnorr-like blind signatures based on modules such as classical groups and lattices. Consequently, our scheme is provably secure in the random oracle model (ROM) against poly-logarithmically-many concurrent sessions assuming the subexponential hardness of the group action inverse problem. In more detail, our blind signature exploits the quadratic twist of an elliptic curve in an essential way to endow isogenies with a strictly richer structure than abstract group actions (but still more restrictive than modules). The basic scheme has public key size 128 B and signature size 8 KB under the CSIDH-512 parameter sets—these are the smallest among all provably secure post-quantum secure blind signatures. Relying on a new ring variant of the group action inverse problem ( rGAIP ), we can halve the signature size to 4 KB while increasing the public key size to 512 B. We provide preliminary cryptanalysis of rGAIP and show that for certain parameter settings, it is essentially as secure as the standard GAIP . Finally, we show a novel way to turn our blind signature into a partially blind signature, where we deviate from prior methods since they require hashing into the set of public keys while hiding the corresponding secret key—constructing such a hash function in the isogeny setting remains an open problem.

For projective Reed–Muller-type codes we give a global duality criterion in terms of the v-number and the Hilbert function of a vanishing ideal. As an application, we provide a global duality theorem for projective Reed–Muller-type codes over Gorenstein vanishing ideals, generalizing the known case where the vanishing ideal is a complete intersection. We classify self dual Reed–Muller-type codes over Gorenstein ideals using the regularity and a parity check matrix. For projective evaluation codes, we give a duality theorem inspired by that of affine evaluation codes. We show how to compute the regularity index of the r -th generalized Hamming weight function in terms of the standard indicator functions of the set of evaluation points.

Circulant Column Parity Mixers (CCPMs) are a particular type of linear maps, used as the mixing layer in permutation-based cryptographic primitives like Keccak - f (SHA3) and Xoodoo . Although being successfully applied, not much is known regarding their algebraic properties. They are limited to invertibility of CCPMs, and that the set of invertible CCPMs forms a group. A possible explanation is due to the complexity of describing CCPMs in terms of linear algebra. In this paper, we introduce a new approach to studying CCPMs using module theory from commutative algebra. We show that many interesting algebraic properties can be deduced using this approach, and that known results regarding CCPMs resurface as trivial consequences of module theoretic concepts. We also show how this approach can be used to study the linear layer of Xoodoo , and other linear maps with a similar structure which we call DCD-compositions. Using this approach, we prove that every DCD-composition where the underlying vector space with the same dimension as that of Xoodoo has a low order. This provides a solid mathematical explanation for the low order of the linear layer of Xoodoo , which equals 32. We design a DCD-composition using this module-theoretic approach, but with a higher order using a different dimension.

To integrate the combination of electric vehicles and wind power of the transmission system, a wind farm environment multi-objective transmission model considering plug-in electric vehicles was established. The fuzzy optimal scheduling model for the environment and economy target of wind farm power system considering grid interconnection of electric vehicles designed by the author has the following characteristics: A fuzzy multi-objective dynamic scheduling model for the design of wind farm power system considering grid connection of plug-in electric vehicle is established, and the original 10 systems are taken as an example for power system design. The results show that the better model shows that the electric vehicle grid connection can greatly reduce the disadvantage of wind power stochastic uncertainties of system scheduling. The model can be used under different conditions further analyzing the rationality of the model determined by the author. An improved particle swarm optimization algorithm is used to solve this problem. The model takes the environmental and economic benefits into account, which improves the efficiency of the model.

Let I be a homogeneous ideal in a polynomial ring S. In this paper, we extend the study of the asymptotic behavior of the minimum distance function δI of I and give bounds for its stabilization point, rI, when I is an F -pure or a square-free monomial ideal. These bounds are related with the dimension and the Castelnuovo–Mumford regularity of I.

In this paper we study the main characteristics of some evaluation codes parameterized by the edges of a bipartite graph with a perfect matching.