Complex Algebra Research Articles

Topology of real multi-affine hypersurfaces and a homological stability property

Let R be a real closed field. We prove that the number of semi-algebraically connected components of a real hypersurface in Rn defined by a multi-affine polynomial of degree d is bounded by 2d−1. This bound is sharp and is independent of n (as opposed to the classical bound of d(2d−1)n−1 on the Betti numbers of hypersurfaces defined by arbitrary polynomials of degree d in Rn due to Petrovskiĭ and Oleĭnik, Thom and Milnor). Moreover, we show there exists c>1, such that given a sequence (Bn)n>0 where Bn is a closed ball in Rn of positive radius, there exist hypersurfaces (Vn⊂Rn)n>0 defined by symmetric multi-affine polynomials of degree 4, such that ∑i⩽5bi(Vn∩Bn)>cn, where bi(⋅) denotes the i-th Betti number with rational coefficients. Finally, as an application of the main result of the paper we verify a representational stability conjecture due to Basu and Riener on the cohomology modules of symmetric real algebraic sets for a new and much larger class of symmetric real algebraic sets than known before.

Matching centroids by a projective transformation

Given two subsets of mathbb {R}^d, when does there exist a projective transformation that maps them to two sets with a common centroid? When is this transformation unique modulo affine transformations? We study these questions for 0- and d-dimensional sets, obtaining several existence and uniqueness results as well as examples of non-existence or non-uniqueness. If both sets have dimension 0, then the problem is related to the analytic center of a polytope and to polarity with respect to an algebraic set. If one set is a single point, and the other is a convex body, then it is equivalent by polar duality to the existence and uniqueness of the Santaló point. For a finite point set versus a ball, it generalizes the Möbius centering of edge-circumscribed convex polytopes and is related to the conformal barycenter of Douady-Earle. If both sets are d-dimensional, then we are led to define the Santaló point of a pair of convex bodies. We prove that the Santaló point of a pair exists and is unique, if one of the bodies is contained within the other and has Hilbert diameter less than a dimension-depending constant. The bound is sharp and is obtained by a box inside a cross-polytope.

Nonnoetherian singularities and their noncommutative blowups

We establish a new fundamental class of varieties in nonnoetherian algebraic geometry related to the central geometry of dimer algebras. Specifically, given an affine algebraic variety X and a finite collection of non-intersecting positive dimensional algebraic sets Y_i \subset X , we construct a nonnoetherian coordinate ring whose variety coincides with X except that each Y_i is identified as a distinct positive dimensional closed point. We then show that the noncommutative blowup of such a singularity is a noncommutative desingularization, in a suitable geometric sense.

First Words, Last Words: New Theories for Reading Old Texts in Sixteenth-Century India. By Yigal Bronner and Lawrence McCrea

In First Words, Last Words, Yigal Bronner and Lawrence McCrea expand on the trending topic of intellectual life in early modern India by discussing in detail the scholastic debate on scriptural interpretation and the role of sequence between the sixteenth-century Dualists, defended by Vyāsatīrtha and Vijayīndratīrtha, and the Nondualists, represented by Appayya Dīkṣita. They present this complex topic with a finesse that will please scholars seasoned in the algebraic complexities of hermeneutics across religious traditions, and with a fluency that can excite academics interested in the intellectual innovativeness of historical South Asia. Their approach focuses on the quest for newness in early modern Indian scholarship which continues to be described as traditionalist even in recent studies that seek to challenge such view (e.g. Pollock 1985, 2001 and Ganeri 2011, 2008; pp. 6–11). By discussing also parallel hermeneutic issues of sequence in Judaism, Christianity, and Islam, they show that social and institutional traditionalising forces were at play in religious traditions across the globe. This contextualisation takes Indic philology out of its too often isolated place. Nevertheless, the discussion from India on its own can fascinate a range of scholars.

The Strong Converse Exponent of Discriminating Infinite-Dimensional Quantum States

The sandwiched Rényi divergences of two finite-dimensional density operators quantify their asymptotic distinguishability in the strong converse domain. This establishes the sandwiched Rényi divergences as the operationally relevant ones among the infinitely many quantum extensions of the classical Rényi divergences for Rényi parameter alpha >1. The known proof of this goes by showing that the sandwiched Rényi divergence coincides with the regularized measured Rényi divergence, which in turn is proved by asymptotic pinching, a fundamentally finite-dimensional technique. Thus, while the notion of the sandwiched Rényi divergences was extended recently to density operators on an infinite-dimensional Hilbert space (in fact, even for states of an arbitrary von Neumann algebra), these quantities were so far lacking an operational interpretation similar to the finite-dimensional case, and it has also been open whether they coincide with the regularized measured Rényi divergences. In this paper we fill this gap by answering both questions in the positive for density operators on an infinite-dimensional Hilbert space, using a simple finite-dimensional approximation technique. We also initiate the study of the sandwiched Rényi divergences, and the related problem of the strong converse exponent, for pairs of positive semi-definite operators that are not necessarily trace-class (this corresponds to considering weights in a general von Neumann algebra setting). This is motivated by the need to define conditional Rényi entropies in the infinite-dimensional setting, while it might also be interesting from the purely mathematical point of view of extending the concept of Rényi (and other) divergences to settings beyond the standard one of positive trace-class operators (positive normal functionals in the von Neumann algebra setting). In this spirit, we also discuss the definition and some properties of the more general family of Rényi (alpha ,z)-divergences of positive semi-definite operators on an infinite-dimensional separable Hilbert space.

Optimal systems, series solutions and conservation laws for a time fractional cancer tumor model

In this paper, we consider certain time-fractional cancer tumor models where the killing rates are space as well as time- dependent. Applying the Lie symmetry analysis, we obtain the infinitesimal transformations and prove that there is a 1-1 and onto mapping between their symmetries. Further, we construct set of Lie algebras and we study the optimality of those algebras. Considering one of the algebras, we obtain similarity variables and under the invariance condition the given partial differential equation with fractional derivative(FPDE) is reduced to ordinary differential equation with fractional order(FODE). Thereafter, we reduce the FODE to the ordinary differential equation(ODE) and present a series solution of the given FPDE with its convergence analysis. Furthermore, we discuss the effect of β on the nature of the solution graphically. Finally, we present the conservation laws for the given model using Lie symmetry.

Edmonds' problem and the membership problem for orbit semigroups of quiver representations

A central problem in algebraic complexity, posed by J. Edmonds [16] , asks to decide if the span of a given l -tuple W = ( W 1 , … , W l ) of N × N complex matrices contains a non-singular matrix. In this paper, we provide a quiver invariant theoretic approach to this problem. Viewing W as a representation of the l -Kronecker quiver K l , Edmonds' problem can be rephrased as asking to decide if there exists a semi-invariant on the representation space ( C N × N ) l of weight ( 1 , − 1 ) that does not vanish at W . In other words, Edmonds' problem is asking to decide if the weight ( 1 , − 1 ) belongs to the orbit semigroup of W . Let Q be an arbitrary acyclic quiver and W a representation of Q . We study the membership problem for the orbit semi-group of W by focusing on the so-called W -saturated weights. We first show that for any given W -saturated weight σ , checking if σ belongs to the orbit semigroup of W can be done in deterministic polynomial time. Next, let ( Q , R ) be an acyclic bound quiver with bound quiver algebra A = K Q / 〈 R 〉 and assume that W satisfies the relations in R . We show that if A / Ann A ( W ) is a tame algebra then any weight σ in the weight semigroup of W is W -saturated. Our results provide a systematic way of producing families of tuples of matrices for which Edmonds' problem can be solved effectively.

Erdős–Ko–Rado and Hilton–Milner theorems for two-forms

The celebrated Erdős–Ko–Rado theorem about the maximal size of an intersecting family of r-element subsets of {1,…,n} was extended to the setting of exterior algebra in [5, Theorem 2.3] and in [6, Theorem 1.4]. However, the equality case has not been settled yet. In this short note, we show that the extension of the Erdős–Ko–Rado theorem and the characterization of the equality case therein, as well as those of the Hilton–Milner theorem to the setting of exterior algebra in the simplest non-trivial case of two-forms follow from a folklore puzzle about possible arrangements of an intersecting family of lines.

Concise tensors of minimal border rank

We determine defining equations for the set of concise tensors of minimal border rank in $${\mathbb {C}}^m{\mathord { \otimes } }{\mathbb {C}}^m{\mathord { \otimes } }{\mathbb {C}}^m$$ when $$m=5$$ and the set of concise minimal border rank $$1_*$$ -generic tensors when $$m=5,6$$ . We solve the classical problem in algebraic complexity theory of classifying minimal border rank tensors in the special case $$m=5$$ . Our proofs utilize two recent developments: the 111-equations defined by Buczyńska–Buczyński and results of Jelisiejew–Šivic on the variety of commuting matrices. We introduce a new algebraic invariant of a concise tensor, its 111-algebra, and exploit it to give a strengthening of Friedland’s normal form for 1-degenerate tensors satisfying Strassen’s equations. We use the 111-algebra to characterize wild minimal border rank tensors and classify them in $${\mathbb {C}}^5{\mathord { \otimes } }{\mathbb {C}}^5{\mathord { \otimes } }{\mathbb {C}}^5$$ .

$$L^\infty $$ estimates for the Banach-valued $${{\bar{\partial }}}$$-problem in a disk

We study the differential equation $$\frac{\partial G}{\partial {{\bar{z}}}}=g$$ with an unbounded Banach-valued Bochner measurable function g on the open unit disk $${\mathbb {D}}\subset {{\mathbb {C}}}$$ . We prove that under some conditions on the growth and essential support of g such equation has a bounded solution given by a continuous linear operator. The obtained results are applicable to the Banach-valued corona problem for the algebra of bounded holomorphic functions on $${\mathbb {D}}$$ with values in a complex commutative unital Banach algebra.

Precisely monotone sets in step-2 rank-3 Carnot algebras

A subset of a Carnot group is said to be precisely monotone if the restriction of its characteristic function to each integral curve of every left-invariant horizontal vector field is monotone. Equivalently, a precisely monotone set is a h-convex set with h-convex complement. Such sets have been introduced and classified in the Heisenberg setting by Cheeger and Kleiner in the 2010’s. In the present paper, we study precisely monotone sets in the wider setting of step-2 Carnot groups, equivalently step-2 Carnot algebras. In addition to general properties, we prove a classification in terms of sublevel sets of h-affine functions in step-2 rank-3 Carnot algebras that can be seen as a generalization of the one obtained by Cheeger and Kleiner in the Heisenberg setting. There is however a significant difference here as it is known that, unlike the Heisenberg setting, there are sublevel sets of h-affine functions on the free step-2 rank-3 Carnot algebra that are not half-spaces.

Finite Algebras for Solid Modeling using Julia’s Sparse Arrays

An early research in solid modeling led by Herbert Voelcker at the University of Rochester and later at Cornell suggested that every solid representation scheme corresponds to an algebra, where the elements of the algebra are solid representations constructed and edited using operations in the algebra. For example, every CSG representation describes an element in a finite Boolean algebra of closed regular sets, whereas every boundary representation describes an element of a vector space of 2-chains in an algebraic topological chain complex. In this paper, we elucidate the precise relationships (functors) between all algebras used for CSG and boundary representations of solids. Based on these properties, we show that many solid modeling operations, including boundary evaluation, reduce to straightforward algebraic operations or application of identified functors that are efficiently implemented using point membership tests and sparse matrix operations. To fully exploit the efficacy of the new algebraic approach to solid modeling, all algorithms are fully implemented in Julia, the modern language of choice for numerical and scientific computing. • New research direction in solid modeling. • Algebraic relationships between CSG and boundary representations of PL solids. • Arrangements are isomorphic to the finite Boolean algebra of regularized sets. • This algebra is isomorphic to a linear vector space of 3-chains. • Operations on CSG and b-reps reduce to composition of algebraic operations. operations.

Algebraic dimension and complex subvarieties of hypercomplex nilmanifolds

A nilmanifold is a (left) quotient of a nilpotent Lie group by a cocompact lattice. A hypercomplex structure on a manifold is a triple of complex structure operators satisfying the quaternionic relations. A hypercomplex nilmanifold is a compact quotient of a nilpotent Lie group equipped with a left-invariant hypercomplex structure. Such a manifold admits a whole 2-dimensional sphere S2 of complex structures induced by quaternions. We prove that for any hypercomplex nilmanifold M and a generic complex structure L∈S2, the complex manifold (M,L) has algebraic dimension 0. A stronger result is proven when the hypercomplex nilmanifold is abelian. Consider the Lie algebra of left-invariant vector fields of Hodge type (1,0) on the corresponding nilpotent Lie group with respect to some complex structure I∈S2. A hypercomplex nilmanifold is called abelian when this Lie algebra is abelian. We prove that all complex subvarieties of (M,L) for generic L∈S2 on a hypercomplex abelian nilmanifold are also hypercomplex nilmanifolds.

The mathcal {F}-Resolvent Equation and Riesz Projectors for the mathcal {F}-Functional Calculus

The Fueter-Sce-Qian mapping theorem gives a constructive way to extend holomorphic functions of one complex variable to slice hyperholomorphic functions. By means of the Cauchy formula for slice hyperholomorphic functions it is possible to have a Fueter-Sce-Qian mapping theorem in integral form for n odd. On this theorem it is based the mathcal {F}-functional calculus for n-tuples of commuting operators. It is a functional calculus based on the commutative version of the S spectrum. Furthermore, it is a monogenic functional calculus in the spirit of McIntosh and collaborators. In this paper, inspired by the quaternionic case and some particular Clifford algebras cases, we show a general resolvent equation for the mathcal {F}-functional calculus in the Clifford algebra setting. Moreover, we prove that the mathcal {F}-resolvent equation is the suitable equation to study the Riesz projectors.

Constructing modular categories from orbifold data

The notion of an orbifold datum \operatorname{\mathbb{A}} in a modular fusion category \mathcal{C} was introduced as part of a generalised orbifold construction for Reshetikhin–Turaev TQFTs by Carqueville, Runkel, and Schaumann in 2018. In this paper, given a simple orbifold datum \operatorname{\mathbb{A}} in \mathcal{C} , we introduce a ribbon category \mathcal{C}_{\operatorname{\mathbb{A}}} and show that it is again a modular fusion category. The definition of \mathcal{C}_{\operatorname{\mathbb{A}}} is motivated by properties of Wilson lines in the generalised orbifold. We analyse two examples in detail: (i) when \operatorname{\mathbb{A}} is given by a simple commutative \Delta -separable Frobenius algebra A in \mathcal{C} ; (ii) when \operatorname{\mathbb{A}} is an orbifold datum in \mathcal{C} = \operatorname{Vect} , built from a spherical fusion category \mathcal{S} . We show that, in case (i), \mathcal{C}_{\operatorname{\mathbb{A}}} is ribbon-equivalent to the category of local modules of A , and, in case (ii), to the Drinfeld centre of \mathcal{S} . The category \mathcal{C}C_{\operatorname{\mathbb{A}}} thus unifies these two constructions into a single algebraic setting.

The search for small association schemes with noncyclotomic eigenvalues

In this article we determine feasible parameter sets for (what could potentially be) commutative association schemes with noncyclotomic eigenvalues that are of smallest possible rank and order. A feasible parameter set for a commutative association scheme corresponds to a standard integral table algebra with integral multiplicities that satisfies all of the parameter restrictions known to hold for association schemes. For each rank and involution type, we generate an algebraic set for which any suitable integral solution corresponds to a standard integral table algebra with integral multiplicities, and then try to find the smallest suitable solution. The main results of this paper show the eigenvalues of association schemes of rank 4 and nonsymmetric association schemes of rank 5 will always be cyclotomic. In the rank 5 cases, the results rely on calculations done by computer for Gröbner bases or for bases of rational vector spaces spanned by polynomials. We give several examples of feasible parameter sets for small symmetric association schemes of rank 5 that have noncyclotomic eigenvalues.

On Lipschitz Geometry at infinity of complex analytic sets

In this article, we study the Lipschitz Geometry at infinity of complex analytic sets and we obtain results on algebraicity of analytic sets and on Bernstein’s problem. Moser’s Bernstein Theorem says that a minimal hypersurface which is a graph of an entire Lipschitz function must be a hyperplane. H. B. Lawson, Jr. and R. Osserman presented examples showing that an analogous result for arbitrary codimension is not true. In this article, we prove a complex parametric version of Moser’s Bernstein Theorem. More precisely, we prove that any entire complex analytic set in $$\mathbb {C}^n$$ which is Lipschitz regular at infinity must be an affine linear subspace of $$\mathbb {C}^n$$ . In particular, a complex analytic set which is a graph of an entire Lipschitz function must be affine linear subspace. That result comes as a consequence of the following characterization of algebraic sets, which is also proved here: if X and Y are entire complex analytic sets which are bi-Lipschitz homeomorphic at infinity then X is a complex algebraic set if and only if Y is a complex algebraic set too. Thus, an entire complex analytic set is a complex algebraic set if and only if it is bi-Lipschitz homeomorphic at infinity to a complex algebraic set. No restrictions on the singular set, dimension nor codimension are required in the results proved here.

Direct Product of Neutrosophic h-ideal in INK-Algebra

In this paper, “we first define the belief of direct product from neutrosophic sets in INK algebras, neutrosophic set, neutrosophic h-ideals, neutrosophic INK-subalgebra and direct product of neutrosophic h-ideals in INK algebras. Let's prove some theorems that show that there is some connection between these principles. Finally, we define the INK subalgebra of the INK algebra and then offer the ideal theorem approximately the connection between its pix and the direct product from the neutrosophic h-ideals.

Data with Intuitionistic Plithogenic attributes and its analysis

Recent time a problem is addressed while dealing the data with multi-valued attributes and its dynamicity. The problem is that these types of data set contain lots of opposite, non-opposite and uncertainty unconsciously. One of the suitable examples is handling soil data sets and its pattern for water storage and plant growth. The precise representation of these types of data sets and its multi-veracity is one of the crucial tasks for the data science researchers. To achieve this goal, algebra of Intuitionistic Plithogenic set and its graph is utilized in this paper. The proposed method is explained using the example of Soil pollution.

On Aupetit’s Scarcity Theorem

Let $ A $ be a complex and unital Banach algebra, $ D $ a domain in $ \mathbb {C} $, and $ f\colon D\to A $ an analytic function. A useful and remarkable result, due to B. Aupetit, is the Scarcity Theorem for elements with finite spectrum; the second part