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Advances in Design Theory, Combinatorial Algebraic Geometry and Applications

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A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Algebra, Geometry and Topology".

Deadline for manuscript submissions: 20 May 2025 | Viewed by 6032

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Special Issue Editors

Prof. Dr. Elena Guardo

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Department of Mathematics and Computer Science, University of Catania, Viale A. Doria, 6, 95100 Catania, Italy
Interests: commutative algebra; algebraic geometry, multilinar algebra; computer algebra; combinatorics; graph theory
Special Issues, Collections and Topics in MDPI journals

Prof. Dr. Mario Gionfriddo

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Dipartimento di Matematica e Informatica, Università di Catania, 95125 Catania, Italy
Interests: combinatorics; graph theory; hypergraphs; block design theory; Steiner systems; number theory; hypergroups
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Prof. Dr. Salvatore Milici

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Department of Mathematics and Computer Science, University of Catania, Catania, Italy
Interests: resolvable decompositions; combinatorics; graph theory; hypergraphs; block design theory; Steiner systems
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Combinatorial algebraic geometry is a branch of mathematics studying objects that can be interpreted from a combinatorial point of view (such as matroids, polytopes, codes, or finite geometries) and also algebraically (using tools from group theory, lattice theory, or commutative algebra), and which has applications in designs, coding theory, cryptography, and number theory.

This Special Issue on “Advances in Combinatorial Algebraic Geometry and Applications” invites front-line researchers and authors to submit original research and review articles on exploring new trends in design theory. It is intended as a bridge between computational issues in the treatment of curves and surfaces (from the symbolic and also numeric points of view) and a combinatorial point of view.

Potential topics include but are not limited to:

Algebraic graph theory;
Finite geometry and designs;
Combinatorial algebraic geometry and its applications;
Combinatorial algebra and its applications;
Coding theory;
Statistical design and experiments;
Cryptography;
Number theory.

Prof. Dr. Elena Guardo
Prof. Dr. Emeritus Mario Gionfriddo
Prof. Dr. Salvatore Milici
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed open access semimonthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

designs
resolvable decompositions
Steiner systems
Hilbert functions
fat points
symbolic and regular powers of ideals: Waldschmidt constant and resurgence

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Published Papers (5 papers)

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Research

34 pages, 1159 KiB

Open AccessFeature PaperArticle

Liftable Point-Line Configurations: Defining Equations and Irreducibility of Associated Matroid and Circuit Varieties

by Oliver Clarke, Giacomo Masiero and Fatemeh Mohammadi

Mathematics 2024, 12(19), 3041; https://doi.org/10.3390/math12193041 - 28 Sep 2024

Viewed by 395

Abstract

We study point-line configurations through the lens of projective geometry and matroid theory. Our focus is on their realization spaces, where we introduce the concepts of liftable and quasi-liftable configurations, exploring cases in which an n-tuple of collinear points can be lifted to a nondegenerate realization of a point-line configuration. We show that forest configurations are liftable and characterize the realization space of liftable configurations as the solution set of certain linear systems of equations. Moreover, we study the Zariski closure of the realization spaces of liftable and quasi-liftable configurations, known as matroid varieties, and establish their irreducibility. Additionally, we compute an irreducible decomposition for their corresponding circuit varieties. Applying these liftability properties, we present a procedure to generate some of the defining equations of the associated matroid varieties. As corollaries, we provide a geometric representation for the defining equations of two specific examples: the quadrilateral set and the

3 \times 4

grid. While the polynomials for the latter were previously computed using specialized algorithms tailored for this configuration, the geometric interpretation of these generators was missing. We compute a minimal generating set for the corresponding ideals. Full article

(This article belongs to the Special Issue Advances in Design Theory, Combinatorial Algebraic Geometry and Applications)

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8 pages, 320 KiB

Open AccessArticle

Uniform {C_h,S(C_h)}-Factorizations of K_n−I for Even h

by Giovanni Lo Faro, Salvatore Milici and Antoinette Tripodi

Mathematics 2023, 11(16), 3479; https://doi.org/10.3390/math11163479 - 11 Aug 2023

Viewed by 1023

Abstract

Let H be a connected subgraph of a graph G. An H-factor of G is a spanning subgraph of G whose components are isomorphic to H. Given a set

H

of mutually non-isomorphic graphs, a uniform

H

-factorization of G [...] Read more.

Let H be a connected subgraph of a graph G. An H-factor of G is a spanning subgraph of G whose components are isomorphic to H. Given a set

H

of mutually non-isomorphic graphs, a uniform

H

-factorization of G is a partition of the edges of G into H-factors for some

H \in H

. In this article, we give a complete solution to the existence problem of uniform

H

-factorizations of

K_{n} - I

(the graph obtained by removing a 1-factor from the complete graph

K_{n}

) for

H = {C_{h}, S (C_{h})}

, where

C_{h}

is a cycle of length an even integer

h \geq 4

and

S (C_{h})

is the graph consisting of the cycle

C_{h}

with a pendant edge attached to each vertex. Full article

(This article belongs to the Special Issue Advances in Design Theory, Combinatorial Algebraic Geometry and Applications)

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31 pages, 1112 KiB

Open AccessArticle

The Newton–Puiseux Algorithm and Triple Points for Plane Curves

by Stefano Canino, Alessandro Gimigliano and Monica Idà

Mathematics 2023, 11(10), 2324; https://doi.org/10.3390/math11102324 - 16 May 2023

Viewed by 1618

Abstract

The paper is an introduction to the use of the classical Newton–Puiseux procedure, oriented towards an algorithmic description of it. This procedure allows to obtain polynomial approximations for parameterizations of branches of an algebraic plane curve at a singular point. We look for an approach that can be easily grasped and is almost self-contained. We illustrate the use of the algorithm first in a completely worked out example of a curve with a point of multiplicity 6, and secondly, in the study of triple points on reduced plane curves. Full article

(This article belongs to the Special Issue Advances in Design Theory, Combinatorial Algebraic Geometry and Applications)

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11 pages, 317 KiB

Open AccessArticle

Recognition and Implementation of Contact Simple Map Germs from (ℂ², 0) → (ℂ², 0)

by Peng Xu, Muhammad Ahsan Binyamin, Adnan Aslam, Muhammad Shahbaz, Saima Aslam and Salma Kanwal

Mathematics 2023, 11(7), 1575; https://doi.org/10.3390/math11071575 - 23 Mar 2023

Cited by 1 | Viewed by 1057

Abstract

The classification of contact simple map germs from

(C^{2}, 0) \to (C^{2}, 0)

was given by Dimca and Gibson. In this article, we give a useful criteria to recognize this classification of contact simple map [...] Read more.

The classification of contact simple map germs from

(C^{2}, 0) \to (C^{2}, 0)

was given by Dimca and Gibson. In this article, we give a useful criteria to recognize this classification of contact simple map germs of holomorphic mappings with finite codimension. The recognition is based on the computation of explicit numerical invariants. By using this characterization, we implement an algorithm to compute the type of the contact simple map germs without computing the normal form and also give its implementation in the computer algebra system Singular. Full article

(This article belongs to the Special Issue Advances in Design Theory, Combinatorial Algebraic Geometry and Applications)

8 pages, 266 KiB

Open AccessArticle

Locally Balanced G-Designs

by Paola Bonacini, Mario Gionfriddo and Lucia Marino

Mathematics 2023, 11(2), 408; https://doi.org/10.3390/math11020408 - 12 Jan 2023

Cited by 1 | Viewed by 1015

Abstract

Let G be a graph and let

K_{n}

be the complete graph of order n. A G-design is a decomposition of the set of edges of

K_{n}

in graphs isomorphic to G, which are called blocks. It [...] Read more.

Let G be a graph and let

K_{n}

be the complete graph of order n. A G-design is a decomposition of the set of edges of

K_{n}

in graphs isomorphic to G, which are called blocks. It is well-known that a G-design is balanced if all the vertices are contained in the number of blocks of G. In this paper, the definition of locally balanced G-design is given, generalizing the existing concepts related to balanced designs. Further, locally balanced G-designs are studied in the cases in which

G ≅ C_{4} + e

and

G ≅ C_{4} + P_{3}

, determining the spectrum. Full article

(This article belongs to the Special Issue Advances in Design Theory, Combinatorial Algebraic Geometry and Applications)

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