We revisit the problem of minimizing a given polynomial f on the hypercube [ − 1 , 1 ] n . Lasserre's hierarchy (also known as the moment- or sum-of-squares hierarchy) provides a sequence of lower bounds { f ( r ) } r ∈ N on the minimum value f ∗ , where r refers to the allowed degrees in the sum-of-squares hierarchy. A natural question is how fast the hierarchy converges as a function of the parameter r. The current state-of-the-art is due to Baldi and Slot [SIAM J. on Applied Algebraic Geometry, 2024] and roughly shows a convergence rate of order 1/r. Here we obtain closely related results via a different approach: the polynomial kernel method. We also discuss limitations of the polynomial kernel method, suggesting a lower bound of order 1 / r 2 for our approach.

We define tropical rational function semifields in n-variables T ( X 1 , … , X n ) ¯ and prove that a tropical curve Γ is realized (except for points at infinity) as the congruence variety V ⊂ R n associated with a congruence on T ( X 1 , … , X n ) ¯ by giving a specific map Γ → V . Also, we shed light on the relation between congruences E on T ( X 1 , … , X n ) ¯ and congruence varieties associated with them and reveal the quotient semifield T ( X 1 , … , X n ) ¯ / E to play the role of coordinate rings that determine isomorphism classes of affine varieties in classical algebraic geometry.

In Algebraic Statistics, M.A. Cueto, J. Morton and B. Sturmfels introduced a statistical model, the Restricted Boltzmann Machine, which introduced the Hadamard product of two or more vectors of an affine or projective space, i.e., the componentwise product of their entries, forcing Algebraic Geometry to enter. The Hadamard product X⋆Y of two subvarieties X,Y⊂Pn is defined as the Zariski closure of the Hadamard product of its elements. Recently, D. Antolini and A. Oneto introduced and studied the definition of Hadamard rank, and we prove some results on it. Moreover, we prove some theorems on the dimension and shape of the Hadamard powers of X. The aim is to describe the images of the Hadamard products without taking the Zariski closure. We also discuss several scenarios describing the case in which some of the data, i.e., the variety X, is wrong or it is not possible to recover it.

We develop a purely combinatorial theory of limit linear series on metric graphs. This will be based on the formalisms of hypercube rank functions and slope structures. We provide a full classification of combinatorial limit linear series of rank one, and discuss connections to other concepts in tropical algebra and combinatorial algebraic geometry.

The concept of viewing graph solvability has gained significant interest in the context of structure-from-motion. A viewing graph is a mathematical structure where nodes are associated with cameras and edges represent the epipolar geometry connecting overlapping views. Solvability studies under which conditions the cameras are uniquely determined by the graph. In this paper we propose a novel framework for analyzing solvability problems based on algebraic geometry, demonstrating its potential in understanding structure-from-motion graphs and proving a conjecture that was previously proposed.

This paper establishes profound connections between metallic ratios, Diophantine equations, and their underlying symmetry groups. Building on recent work on Diophantine equations and their solutions, we develop a comprehensive theory revealing the algebraic and geometric structures governing families of Diophantine equations associated with metallic ratios. Through detailed investigations of automorphism groups, continued fractions, and arithmetic geometry, we provide complete proofs and explicit examples that illuminate the deep arithmetic properties of metallic ratios. Our work offers a unified framework bridging number theory, group theory, and algebraic geometry, with applications to class field theory and computational number theory.

The objective of this review article is to synthesize and present, in a detailed manner, the technique of specialization of curves. Specifically, we review and analyze key examples on how certain curves of degree 4 and arithmetic genus 0 (parametrized by the Hilbert scheme Hilb4m+1(ℙ3)) can transform or degenerate into other curves. The computational calculation procedure is detailed and illustrated through the use of the Macaulay2 software. Finally, the utility of specializations for explicitly describing and classifying the irreducible components of Hilbert schemes and their corresponding Chow varieties is evaluated. The technique of specialization of families of curves has established itself as a powerful and rigorous method in algebraic geometry. By providing a well-defined stratification diagram, this tool allows for a precise classification of curves and an understanding of the connectivity of the components of the Hilbert scheme.

This study aims to develop a mathematics educational game for seventh-grade junior high school students using Construct 2 software, focusing on algebra and plane geometry. The research method used is experimental, covering needs analysis, system design, implementation, and black box testing. The game is designed with interactive features such as multiple-choice quizzes, visual simulations, and score tracking to enhance motivation and conceptual understanding. A questionnaire completed by 15 respondents showed that 86.7% agreed/strongly agreed the game interface was easy to use, 80% felt the game supported concept comprehension, and 93.3% would recommend the game. Functional testing confirmed that all features (navigation, content, quizzes, and score storage) operated optimally. The game aligns with the Merdeka Curriculum, which emphasizes competency-based and contextual learning. The results conclude that the Construct 2-based educational game is effective as an interactive learning tool, increases student engagement, and is suitable for further development by adding collaborative features and expanding its content.

Abstract In the wake of independence, impelled by Nehru’s vision, India saw a great flowering of scientific institutions. Among these was the Tata Institute of Fundamental Research (TIFR), and this is where M. S. Narasimhan grew from being a graduate student to a towering figure in Indian science. He distinguished himself as a mathematician, teacher and administrator. After his retirement from TIFR, he went on to a second innings as director of the Mathematics Department at the International Centre for Theoretical Physics in Trieste, where he built up a strong school in algebraic geometry. Narasimhan was an extraordinarily versatile researcher, contributing significantly to analysis, representation theory, differential geometry and algebraic geometry. Two related discoveries have in particular proved to be of great importance. The first is the application of stability in classifying a family of algebro-geometric objects, identifying the dominant part that parameterizes semistable objects, and then adding other components built out of extensions of smaller semistable objects. The second insight is that stable objects are characterized as those that satisfy non-linear partial differential equations, a discovery that foretold major developments 20 years later.

Abstract This article presents geometric optimal control techniques for analyzing geodesics in time-optimal Zermelo navigation problems on 2-spheres of revolution. We classify the problem by analyzing the pair ( F 0 , g ) , which represents the current (or wind) and the Riemannian metric. Using the maximum principle, the dynamics of geodesics are described by a Hamiltonian vector field on the cotangent bundle T ∗ S 2 . Our primary motivation is the application to micromagnetism, specifically spin magnetization reversal in ferromagnetic ellipsoidal samples. This model depends on four parameters and the amplitude of the applied magnetic field. The problem is formulated as a Zermelo navigation on the 2-sphere, where geodesics are classified as elliptic, hyperbolic, or abnormal. We demonstrate that the transition set | F 0 | g = 1 , which separates weak and strong current domains, is critical for understanding optimality. A key result shows that abnormal geodesics intersect this set with semi-cubical cusp singularities, a phenomenon we term the Landau–Lifshitz billiard. The analysis of the transition set’s connected components is complex and complemented by algebraic geometry and symbolic computations. We further reveal that hyperbolic geodesics lose optimality at their second intersection with the abnormal arc. Our numerical simulations complement this analysis by computing conjugate and cut loci, wavefronts, and accessibility sets, providing new insights into optimal magnetization switching under bounded control.

In the forty years of existence of derived category, it was first thought as a tool in algebraic geometry, especially in the development of duality theories that were done by Hartshorne and others. After these first moments, the theory provided a powerful homological tool for the study of linear differential equations. The basic example in the literature that can be found about this is the Riemann-Hilbert problem of associating suitable regular systems of differential equations to constructible sheaves. This studies can be found in the work of Kashiwara and Schapira. See M. Kashiwara, P. Schapira “Sheaves on manifolds" ([15]). To understand the structure of the derived category is necessary to study the axioms of triangulated categories that were introduced in the mid 1960’s by J.L.Verdier in his thesis “Des catégories dérivées des catégories abéliennes" ([21]). The role of the triangles in the derived category is a similar role of the exact sequence in the abelian category. But it is important to remember that these axioms had their origins in algebraic geometry and algebraic topology. Nowadays there are important applications of triangulated categories in areas like algebraic geometry, algebraic topology (stable homotopy theory), commutative algebra, differential geometry and representation theory of artin algebras. See, for instance, the book of D. Happel- “Triangulated categories and the representation theory of finite dimensional algebras" ([11]). The objective of this notes is to present an introdutory material to the undergraduate and graduate students that would like to know some ideas about the derived category. These are the notes a one week series of introductory lectures which I gave in the XXIII-Escola de Algebra, in Maringá, Paraná, Brazil. Firstly we introduced the concepts of additive and abelian category to show the axioms of triangulated category that are our main objective. The triangulated category obey four axioms. We first introduced the first three axioms and their consequences on chapter one and then the octahedral axioms in various equivalent forms in a separate section of the first chapter. The objective of this section is to give a model capable of making this axiom more palatable since, in general, the form that it is presented in the literature does not remind the reader of any similar structure in other fields of mathematics. So, we make the necessary efforts here to present another form of this axiom that is similar to other tools that could be seen in the abelian categories. We present in chapter one the main example of triangulated category, the homotopy category of complexes. Secondly, to understand the morphisms in the derived category I introduced the concept of localization in chapter two. To those that are starting to study localization, we present the necessary background to understand the localization of non commutative ring. We believe that with this model in mind the student will profit more from the study of localization of categories. On chapter two, the student will find the necessary information and exercises to begin to manipulate morphisms in the derived category. So, on chapter three we introduce the definition of derived category of an abelian category and we explain how one sees the original abelian category as a subcategory of its derived category. After having done all this work, it is natural to have many questions about the behavior of the derived category or its applications. Therefore, we present here a bibliography in portuguese and in english that will help the students to make further investigations. The reader that whishes to know the history and the motivation of the begining of the derived category with many details, should read the introduction of the book "Sheaves on Manifolds - M. Kashiwara and M. Schapira ([15]). Acknowledgements: I am particularly grateful to Sônia Maria Fernandes-DMAUFV, Tanise Carnieri Pierin -DMAT-UFPR and Eduardo Nascimento Marcos IMEUSP, who carefully worked through the text and sent me detailed lists of corrections, questions and remarks. These notes were writen for the first time in 2014 and were used in a minicourse which I tough in the XXIII-Escola de Algebra in Maringá, Paraná, Brazil. The last version was written during my visit to IME-USP in 2018, where I got finantial help of Fapesp, process 2018/08104 - 3.

Solving quadratics with compass: The geometry of Algebra

Abstract We introduce a geometric model of shallow multiplicative exponential linear logic (MELL) using the Hilbert scheme. Building on previous work interpreting multiplicative linear logic (MLL) proofs as systems of linear equations, we show that shallow MELL proofs can be modelled by locally projective schemes. The key insight is that while MLL proofs correspond to equations between formulas, the exponential fragment of shallow proofs corresponds to equations between these equations. We prove that the model is invariant under cut-elimination by constructing explicit isomorphisms between the schemes associated with proofs related by cut-reduction steps. A key technical tool is the interpretation of the exponential modality using the Hilbert scheme, which parameterises closed subschemes of projective space. We demonstrate the model through detailed examples, including an analysis of Church numerals that reveals how the Hilbert scheme captures the geometric content of promoted formulas. This work establishes new connections between proof theory and algebraic geometry, suggesting broader relationships between computation and scheme theory.

Many discrete mathematics models emphasize the Catalan numbers, a series of natural integers important in combinatory. These numbers also appear in lattice routes, binary bushes, and create triangulations. This study uses combinatorial methods to organize discrete arithmetic models using Catalan numbers. It emphasizes algebraic combinatory, geometry, and plan idea. We show how Catalan numbers regulate combinatorial devices and use them to create form analysis algorithms and solve optimization problems. We demonstrate how they may be used to count optimal search tree topologies, non-crossing partitions, and planar graphs. We also provide unique approaches to extend Catalan-based models to higher dimensions, increasing their relevance in modern mathematical modelling.

This study investigates the intersection of algebraic geometry and coding theory, specifically focusing on the application of algebraic curves in the advancement of error-correcting codes. Algebraic curves, as mathematical objects, offer profound implications in the design and analysis of error-correcting codes, providing robust solutions to the challengesof data transmission and storage. This paper delves into the theoretical foundations ofalgebraic curves, their role in constructing powerful error-correcting codes, and the practicalapplications of these codes in various technological domains.

In this memoir, we study the counterpart of Grothendieck’s projectivization construction in the context of derived algebraic geometry. Our main results are as follows: First, we define the derived projectivization of a connective complex, study its fundamental properties such as finiteness properties and functorial behaviors, and provide explicit descriptions of their relative cotangent complexes. We then focus on the derived projectivizations of complexes of perfect-amplitude contained in [ 0 , 1 ] [0, 1] . In this case, we prove a generalized Serre’s theorem, a derived version of Beilinson’s relations, and establish semiorthogonal decompositions for their derived categories. Finally, we show that many moduli problems fit into the framework of derived projectivizations, such as moduli spaces that arise in Hecke correspondences. We apply our results to these situations.

We propose an algebraic geometry-inspired approach for constructing entangled subspaces within the Hilbert space of a multipartite quantum system. Specifically, our method employs a modified Veronese embedding, restricted to the conic, to define subspaces within the symmetric part of the Hilbert space. By utilizing this technique, we construct the minimal-dimensional, non-orthogonal yet Unextendible Product Basis (nUPB), enabling the decomposition of the multipartite Hilbert space into a two-dimensional subspace, complemented by a Genuinely Entangled Subspace (GES) and a maximal-dimensional Completely Entangled Subspace (CES). In multiqudit systems, we determine the maximum achievable dimension of a symmetric GES and demonstrate its realization through this construction. Furthermore, we systematically investigate the transition from the conventional Veronese embedding to the modified one by imposing various constraints on the affine coordinates, which, in turn, increases the CES dimension while reducing that of the GES.

We study the algebraic complexity of Euclidean Distance (ED) minimization from a generic tensor to a variety of rank-one tensors. The ED degree of the Segre-Veronese variety counts the number of complex critical points of this optimization problem. We regard this invariant as a function of inner products. We prove that Frobenius inner product is a local minimum of the ED degree, and conjecture that it is a global minimum. We prove our conjecture in the case of matrices and symmetric binary and 3 × 3 × 3 3\times 3\times 3 tensors. We discuss the above optimization problem for other algebraic varieties, classifying all possible values of the ED degree. Our approach combines tools from Singularity Theory, Morse Theory, and Algebraic Geometry.

It is an important foundational result in algebraic geometry that the 2 2 -category of quasi-coherent modules fibered over the category of commutative rings satisfies descent with respect to the fqpc topology. More recently, Akhil Mathew and Jacob Lurie introduced an ∞ \infty -categorical version of this result for module spectra using the notion of descendable E ∞ \mathbb {E}_{\infty } -ring maps . In this paper, we will construct a faithfully flat algebra over the infinite polynomial ring on an algebraically closed field that is not descendable.

This paper presents some nontrivial computational results on derived category and Fourier–Mukai technique in algebraic geometry. In particular, it aims at presenting calculations involving spherical twists as a certain class of Fourier–Mukai functors and its cohomological descent on the singular rational cohomology of smooth projective variety. The purpose of this investigation is to present a new perspective, based upon Fourier–Mukai technique, on solving classical problems involving characteristic classes: in particular, the Chern and the Euler characteristics.