Genericity Assumptions Research Articles

On the operators with numerical range in an ellipse

We give new necessary and sufficient conditions for the numerical range W(T) of an operator T∈B(H) to be a subset of the closed elliptical set Kδ⊆C given byKδ=def{x+iy:x2(1+δ)2+y2(1−δ)2≤1}, where 0<δ<1. Here B(H) denotes the collection of bounded linear operators on a Hilbert space H. Central to our efforts is a direct generalization of Berger's well-known criterion for an operator to have numerical radius at most one, his so-called strange dilation theorem. Specifically, we show that, if T acts on a finite-dimensional Hilbert space H and satisfies a certain genericity assumption, then W(T)⊆Kδ if and only if there exists a Hilbert space K⊇H, operators X1 and X2 on H and a unitary operator U acting on K such that(0.1)X1+X2=T,X1X2=δ and(0.2)X1k+X2k=2PHUk|H,k=1,2,…, where PH denotes the orthogonal projection from K to H.We next generalize the lemma of Sarason that describes power dilations in terms of semi-invariant subspaces to operators T that satisfy the relations (0.1) and (0.2). This generalization yields a characterization of the operators T∈B(H) such that W(T) is contained in Kδ in terms of certain structured contractions that act on H⊕H.As a corollary of our results we extend Ando's parametrization of operators having numerical range in a disc to those T such that W(T)⊆Kδ. We prove that, if T acts on a finite-dimensional Hilbert space H, then W(T)⊆Kδ if and only if there exist a pair of contractions A,B∈B(H) such that A is self-adjoint andT=2δA+(1−δ)1+AB1−A. We also obtain a formula for the B. and F. Delyon calcular norm of an analytic function on the inside of an ellipse in terms of the extremal H∞-extension problem for analytic functions defined on a slice of the symmetrized bidisc.

Dimension results for extremal-generic polynomial systems over complete toric varieties

We study polynomial systems with prescribed monomial supports in the Cox ring of a toric variety built from a complete polyhedral fan. We present combinatorial formulas for the dimension of their associated subvarieties under genericity assumptions on the coefficients of the polynomials. Using these formulas, we identify at which degrees generic systems in polytopal algebras form regular sequences. Our motivation comes from sparse elimination theory, where knowing the expected dimension of these subvarieties leads to specialized algorithms and to large speed-ups for solving sparse polynomial systems. As a special case, we classify the degrees at which regular sequences defined by weighted homogeneous polynomials can be found, answering an open question in the Gröbner bases literature. We also show that deciding whether a sparse system is generically a regular sequence in a polytopal algebra is hard from the point of view of theoretical computational complexity.

Cosmology from random entanglement

We construct entangled microstates of a pair of holographic CFTs whose dual semiclassical description includes big bang-big crunch AdS cosmologies in spaces without boundaries. The cosmology is supported by inhomogeneous heavy matter and it partially purifies the bulk entanglement of two disconnected auxiliary AdS spacetimes. We show that the island formula for the fine grained entropy of one of the CFTs follows from a standard gravitational replica trick calculation. In generic settings, the cosmology is contained in the entanglement wedge of one of the two CFTs. We then investigate properties of the cosmology-to-boundary encoding map, and in particular, its non-isometric character. Restricting our attention to a specific class of states on the cosmology, we provide an explicit, and state-dependent, boundary representation of operators acting on the cosmology. Finally, under genericity assumptions, we argue for a non-isometric to approximately-isometric transition of the cosmology-to-boundary map for “simple” states on the cosmology as a function of the bulk entanglement, with tensor network toy models of our setup as a guide.

On the mod p cohomology for GL2

Let p be a prime number and F a totally real number field unramified at places above p. Let r‾:Gal(F‾/F)→GL2(Fp‾) be a modular Galois representation which satisfies the Taylor-Wiles hypothesis and some technical genericity assumptions. For v a fixed place of F above p, we prove that many of the admissible smooth representations of GL2(Fv) over Fp‾ associated to r‾ in the corresponding Hecke-eigenspaces of the mod p cohomology have Gelfand–Kirillov dimension [Fv:Qp]. This builds on and extends the work of Breuil-Herzig-Hu-Morra-Schraen in [2] and Hu-Wang in [12], giving a unified proof in all cases (r‾ either semisimple or not at v).

On the Polyhedral Homotopy Method for Solving Generalized Nash Equilibrium Problems of Polynomials

The generalized Nash equilibrium problem (GNEP) is a kind of game to find strategies for a group of players such that each player’s objective function is optimized. Solutions for GNEPs are called generalized Nash equilibria (GNEs). In this paper, we propose a numerical method for finding GNEs of GNEPs of polynomials based on the polyhedral homotopy continuation and the Moment-SOS hierarchy of semidefinite relaxations. We show that our method can find all GNEs if they exist, or detect the nonexistence of GNEs, under some genericity assumptions. Some numerical experiments are made to demonstrate the efficiency of our method.

On the expressiveness of Lara: A proposal for unifying linear and relational algebra

We study the expressive power of the Lara language – a recently proposed unified model for expressing relational and linear algebra operations – both in terms of traditional database query languages and some analytic tasks often performed in machine learning pipelines. Since Lara is parameterized by a set of user-defined functions which allow to transform values in tables, known as extension functions, the exact expressive power of the language depends on how these functions are defined. We start by showing Lara to be expressive complete with respect to a syntactic fragment of relational algebra with aggregation (under the mild assumption that extension functions in Lara can cope with traditional relational algebra operations such as selection and renaming). We then look further into the expressiveness of Lara based on different classes of extension functions, and distinguish two main cases depending on the level of genericity that queries are enforced to satisfy. Under strong genericity assumptions the language cannot express matrix convolution, a very important operation in current machine learning pipelines. This language is also local, and thus cannot express operations such as matrix inverse that exhibit a recursive behavior. For expressing convolution, one can relax the genericity requirement by adding an underlying linear order on the domain. This, however, destroys locality and turns the expressive power of the language much more difficult to understand. In particular, although under complexity assumptions some versions of the resulting language can still not express matrix inverse, a proof of this fact without such assumptions seems challenging to obtain.

The numerical algebraic geometry of bottlenecks

This is a computational study of bottlenecks on algebraic varieties. The bottlenecks of a smooth variety X⊆Cn are the lines in Cn which are normal to X at two distinct points. The main result is a numerical homotopy that can be used to approximate all isolated bottlenecks. This homotopy has the optimal number of paths under certain genericity assumptions. In the process we prove bounds on the number of bottlenecks in terms of the Euclidean distance degree. Applications include the optimization problem of computing the distance between two real varieties. Also, computing bottlenecks may be seen as part of the problem of computing the reach of a smooth real variety and efficient methods to compute the reach are still to be developed. Relations to triangulation of real varieties and meshing algorithms used in computer graphics are discussed in the paper. The resulting algorithms have been implemented with Bertini [4] and Macaulay2 [17].

Symmetric powers of algebraic and tropical curves: A non-Archimedean perspective

We show that the non-Archimedean skeleton of the d d -th symmetric power of a smooth projective algebraic curve X X is naturally isomorphic to the d d -th symmetric power of the tropical curve that arises as the non-Archimedean skeleton of X X . The retraction to the skeleton is precisely the specialization map for divisors. Moreover, we show that the process of tropicalization naturally commutes with the diagonal morphisms and the Abel-Jacobi map and we exhibit a faithful tropicalization for symmetric powers of curves. Finally, we prove a version of the Bieri-Groves Theorem that allows us, under certain tropical genericity assumptions, to deduce a new tropical Riemann-Roch-Theorem for the tropicalization of linear systems.

Equivariant stable sheaves and toric GIT

For $(X,\,L)$ a polarized toric variety and $G\subset \mathrm {Aut}(X,\,L)$ a torus, denote by $Y$ the GIT quotient $X/\!\!/G$. We define a family of fully faithful functors from the category of torus equivariant reflexive sheaves on $Y$ to the category of torus equivariant reflexive sheaves on $X$. We show, under a genericity assumption on $G$, that slope stability is preserved by these functors if and only if the pair $((X,\,L),\,G)$ satisfies a combinatorial criterion. As an application, when $(X,\,L)$ is a polarized toric orbifold of dimension $n$, we relate stable equivariant reflexive sheaves on certain $(n-1)$-dimensional weighted projective spaces to stable equivariant reflexive sheaves on $(X,\,L)$.

A generalization of Zakalyukin's lemma, and symmetries of surface singularities

Zakalyukin's lemma asserts that the coincidence of the images of two wave front germs implies the right equivalence of corresponding map germs under a certain genericity assumption. The purpose of this paper is to give an improvement of this lemma for frontals. Moreover, we give several applications for singularities on surfaces.

Solving parametric systems of polynomial equations over the reals through Hermite matrices

We design a new algorithm for solving parametric systems of equations having finitely many complex solutions for generic values of the parameters. More precisely, let f=(f1,…,fm)⊂Q[y][x] with y=(y1,…,yt) and x=(x1,…,xn), V⊂Ct×Cn be the algebraic set defined by the simultaneous vanishing of the fi's and π be the projection (y,x)→y. Under the assumptions that f admits finitely many complex solutions when specializing y to generic values and that the ideal generated by f is radical, we solve the following algorithmic problem. On input f, we compute semi-algebraic formulas defining open semi-algebraic sets S1,…,Sℓ in the parameters' space Rt such that ∪i=1ℓSi is dense in Rt and, for 1≤i≤ℓ, the number of real points in V∩π−1(η) is invariant when η ranges over Si.This algorithm exploits special properties of some well chosen monomial bases in the quotient algebra Q(y)[x]/I where I⊂Q(y)[x] is the ideal generated by f in Q(y)[x] as well as the specialization property of the so-called Hermite matrices which represent Hermite's quadratic forms. This allows us to obtain “compact” representations of the semi-algebraic sets Si by means of semi-algebraic formulas encoding the signature of a given symmetric matrix.When f satisfies extra genericity assumptions (such as regularity), we use the theory of Gröbner bases to derive complexity bounds both on the number of arithmetic operations in Q and the degree of the output polynomials. More precisely, letting d be the maximal degrees of the fi's and D=n(d−1)dn, we prove that, on a generic input f=(f1,…,fn), one can compute those semi-algebraic formulas using O˜((t+Dt)23tn2t+1d3nt+2(n+t)+1) arithmetic operations in Q and that the polynomials involved in these formulas have degree bounded by D.We report on practical experiments which illustrate the efficiency of this algorithm, both on generic parametric systems and parametric systems coming from applications since it allows us to solve systems which were out of reach on the current state-of-the-art.

Interconnection, Reciprocity and a Hierarchical Classification of Generalized Multiports

Interconnection of (linear, time-invariant, resistive) n-ports is a fundamental concept of circuit theory. We discuss their full and terminal interconnectability, analogously to full and terminal solvability of networks and provide a necessary condition for the full interconnectability (also sufficient under some genericity assumption), which can be checked in polynomial time using matroid algorithms. Reciprocity is another fundamental concept for ordinary n-ports, where the number r of linearly independent equations equals to n, There is a well-known concept of generalized or g-reciprocity if r > n. For r <; n we introduce the concept of potential or p-reciprocity if further equations can be added to the equation system of the n-port to obtain an ordinary reciprocal n-port. We prove that an n-port is p-reciprocal if and only if its dual is g-reciprocal and study the relations of these concepts to the spectrum of the n-port, leading to a hierarchical classification of n-ports. We also prove that the interconnection of p- or g-reciprocal n-ports preserves these properties, give polynomial algorithms to recognize p- or g-reciprocity and, in case of p-reciprocity, to extend the equation system to that of a reciprocal n-port.

Computing 𝐿-polynomials of Picard curves from Cartier–Manin matrices

We study the sequence of zeta functions Z ( C p , T ) Z(C_p,T) of a generic Picard curve C : y 3 = f ( x ) C:y^3=f(x) defined over Q \mathbb {Q} at primes p p of good reduction for C C . We define a degree 9 polynomial ψ f ∈ Q [ x ] \psi _f\in \mathbb {Q}[x] such that the splitting field of ψ f ( x 3 / 2 ) \psi _f(x^3/2) is the 2 2 -torsion field of the Jacobian of C C . We prove that, for all but a density zero subset of primes, the zeta function Z ( C p , T ) Z(C_p,T) is uniquely determined by the Cartier–Manin matrix A p A_p of C C modulo p p and the splitting behavior modulo p p of f f and ψ f \psi _f ; we also show that for primes ≡ 1 ( mod 3 ) \equiv 1 \pmod {3} the matrix A p A_p suffices and that for primes ≡ 2 ( mod 3 ) \equiv 2 \pmod {3} the genericity assumption on C C is unnecessary. An element of the proof, which may be of independent interest, is the determination of the density of the set of primes of ordinary reduction for a generic Picard curve. By combining this with recent work of Sutherland, we obtain a practical deterministic algorithm that computes Z ( C p , T ) Z(C_p,T) for almost all primes p ≤ N p \le N using N log ⁡ ( N ) 3 + o ( 1 ) N\log (N)^{3+o(1)} bit operations. This is the first practical result of this type for curves of genus greater than 2.

The Saddle Point Problem of Polynomials

This paper studies the saddle point problem of polynomials. We give an algorithm for computing saddle points. It is based on solving Lasserre’s hierarchy of semidefinite relaxations. Under some genericity assumptions on defining polynomials, we show that: (i) if there exists a saddle point, our algorithm can get one by solving a finite hierarchy of Lasserre-type semidefinite relaxations; (ii) if there is no saddle point, our algorithm can detect its nonexistence.

Deterministic computation of the characteristic polynomial in the time of matrix multiplication

This paper describes an algorithm which computes the characteristic polynomial of a matrix over a field within the same asymptotic complexity, up to constant factors, as the multiplication of two square matrices. Previously, this was only achieved by resorting to genericity assumptions or randomization techniques, while the best known complexity bound with a general deterministic algorithm was obtained by Keller-Gehrig in 1985 and involves logarithmic factors. Our algorithm computes more generally the determinant of a univariate polynomial matrix in reduced form, and relies on new subroutines for transforming shifted reduced matrices into shifted weak Popov matrices, and shifted weak Popov matrices into shifted Popov matrices.

Can the Genericity Assumption Decrease the Rank of a Matrix?

The genericity assumption, supposing that the nonzero parameters of a system are algebraically independent transcendentals over the field of the rationals, often helps for the mathematical modelling of linear systems. Without this condition nonzero expansion members of a determinant can cancel out each other, decreasing the rank of a matrix. In this note we show that under some circumstances an increase is also possible. This counterintuitive phenomenon is explained using some tools from matroid theory, and is illustrated by a classical network of Carlin and Youla.

Computing symmetric determinantal representations

We introduce the DeterminantalRepresentations package for Macaulay2, which computes definite symmetric determinantal representations of real polynomials. We focus on quadrics and plane curves of low degree (i.e. cubics and quartics). Our algorithms are geared towards speed and robustness, employing linear algebra and numerical algebraic geometry, without genericity assumptions on the polynomials.

Effective coefficient asymptotics of multivariate rational functions via semi-numerical algorithms for polynomial systems

The coefficient sequences of multivariate rational functions appear in many areas of combinatorics. Their diagonal coefficient sequences enjoy nice arithmetic and asymptotic properties, and the field of analytic combinatorics in several variables (ACSV) makes it possible to compute asymptotic expansions. We consider these methods from the point of view of effectivity. In particular, given a rational function, ACSV requires one to determine a (generically) finite collection of points that are called critical and minimal. Criticality is an algebraic condition, meaning it is well treated by classical methods in computer algebra, while minimality is a semi-algebraic condition describing points on the boundary of the domain of convergence of a multivariate power series. We show how to obtain dominant asymptotics for the diagonal coefficient sequence of multivariate rational functions under some genericity assumptions using symbolic-numeric techniques. To our knowledge, this is the first completely automatic treatment and complexity analysis for the asymptotic enumeration of rational functions in an arbitrary number of variables.

Min-max embedded geodesic lines in asymptotically conical surfaces

We employ min-max methods to construct uncountably many, geometrically distinct, properly embedded geodesic lines in any asymptotically conical surface of non-negative scalar curvature, a setting where minimization schemes are doomed to fail. Our construction provides control of the Morse index of the geodesic lines we produce, which will be always less or equal than one (with equality under suitable curvature or genericity assumptions), as well as of their precise asymptotic behaviour. In fact, we can prove that in any such surface for every couple of opposite half-lines there exists an embedded geodesic line whose two ends are asymptotic, in a suitable sense, to those half-lines.

Towards the Finite Slope Part for GLn

Abstract Let $L$ be a finite extension of ${\mathbb{Q}}_p$ and $n\geq 2$. We associate to a crystabelline $n$-dimensional representation of ${\operatorname{Gal}}(\overline L/L)$ satisfying mild genericity assumptions a finite length locally ${\mathbb{Q}}_p$-analytic representation of ${\operatorname{GL}}_n(L)$. In the crystalline case and in a global context, using the recent results on the locally analytic socle from [6], we prove that this representation indeed occurs in spaces of $p$-adic automorphic forms. We then use this latter result in the ordinary case to show that certain “ordinary” $p$-adic Banach space representations constructed in our previous work appear in spaces of $p$-adic automorphic forms. This gives strong new evidence to our previous conjecture in the $p$-adic case.