| 1. |
- Jost, Christine, 1984-
(author)
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Topics in Computational Algebraic Geometry and Deformation Quantization
- 2013
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Doctoral thesis (other academic/artistic)abstract
- This thesis consists of two parts, a first part on computations in algebraic geometry, and a second part on deformation quantization. More specifically, it is a collection of four papers. In the papers I, II and III, we present algorithms and an implementation for the computation of degrees of characteristic classes in algebraic geometry. Paper IV is a contribution to the field of deformation quantization and actions of the Grothendieck-Teichmüller group.In Paper I, we present an algorithm for the computation of degrees of Segre classes of closed subschemes of complex projective space. The algorithm is based on the residual intersection theorem and can be implemented both symbolically and numerically.In Paper II, we describe an algorithm for the computation of the degrees of Chern-Schwartz-MacPherson classes and the topological Euler characteristic of closed subschemes of complex projective space, provided an algorithm for the computation of degrees of Segre classes. We also explain in detail how the algorithm in Paper I can be implemented numerically. Together this yields a symbolical and a numerical version of the algorithm.Paper III describes the Macaulay2 package CharacteristicClasses. It implements the algorithms from papers I and II, as well as an algorithm for the computation of degrees of Chern classes.In Paper IV, we show that L-infinity-automorphisms of the Schouten algebra T_poly(R^d) of polyvector fields on affine space R^d which satisfy certain conditions can be globalized. This means that from a given L-infinity-automorphism of T_poly(R^d) an L-infinity-automorphism of T_poly(M) can be constructed, for a general smooth manifold M. It follows that Willwacher's action of the Grothendieck-Teichmüller group on T_poly(R^d) can be globalized, i.e., the Grothendieck-Teichmüller group acts on the Schouten algebra T_poly(M) of polyvector fields on a general manifold M.
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| 2. |
- Ndikubwayo, Innocent, 1986-
(author)
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Polynomial Sequences Generated by Linear Recurrences : Location and Reality of Zeros
- 2021
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Doctoral thesis (other academic/artistic)abstract
- In this thesis, we study the problem of location of the zeros of individual polynomials in sequences of polynomials generated by linear recurrence relations.In paper I, we establish the necessary and sufficient conditions that guarantee hyperbolicity of all the polynomials generated by a three-term recurrence of length 2, whose coefficients are arbitrary real polynomials. These zeros are dense on the real intervals of an explicitly defined real semialgebraic curve.Paper II extends Paper I to three-term recurrences of length greater than 2. We prove that there always exist non-hyperbolic polynomial(s) in the generated sequence. We further show that with at most finitely many known exceptions, all the zeros of all the polynomials generated by the recurrence lie and are dense on an explicitly defined real semialgebraic curve which consists of real intervals and non-real segments. The boundary points of this curve form a subset of zero locus of the discriminant of the characteristic polynomial of the recurrence.Paper III discusses the zero set for polynomials generated by three-term recurrences of lengths 3 and 4 with arbitrary polynomial coefficients. We prove that except the zeros of the polynomial coefficients, all the zeros of every generated polynomial lie on an explicitly defined real semialgebraic curve.Paper IV extends the results in paper III and generalizes a conjecture by K. Tran [2]. We consider a three-term recurrence relation of any length whose coefficients are arbitrary complex polynomials and prove that with the exception of the zeros of the polynomial coefficients, all the zeros of every generated polynomial lie on a real algebraic curve. We derive the equation of this curve.Paper V establishes the necessary and sufficient conditions guaranteeing the reality of all the zeros of every polynomial generated by a special five-term recurrence with real coefficients. We put the problem in the context of banded Toeplitz matrices whose associated Laurent polynomial is holomorphic in the punctured plane. We interpret the conditions in terms of the positivity/negativity of the discriminant of a certain polynomial whose coefficients are explicit functions of the parameters in the recurrence.
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| 3. |
- Oneto, Alessandro, 1988-
(author)
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Power ideals, Fröberg conjecture and Waring problems
- 2014
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Licentiate thesis (other academic/artistic)abstract
- This thesis is divided into two chapters. First, we want to study particularclasses of power ideals, with particular attention to their relation with the Fröberg conjecture on the Hilbert series of generic ideals. In the second part,we study a generalization (introduced by Fröberg, Ottaviani, and Shapiro in 2012)of the classical Waring problem for polynomials about writing homogeneouspolynomials as sums of powers. We see also how the theories of fat points andsecant varieties of Veronese varieties play a crucial role in the relation betweenthose chapters and in providing tools to nd an answer to our questions.The main results are the computation of the Hilbert series of particularclasses of power ideals, which in particular give us a proof of the Fröberg conjecturefor generic ideals generated by eight homogeneous polynomials of thesame degree in four variables, and the solution of the generalized Waring problemin the case of sums of squares in three and four variables. We also beginthe study of the generalized Waring problem for monomials.
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| 4. |
- Alexandersson, Per, 1987-
(author)
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Combinatorial Methods in Complex Analysis
- 2013
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Doctoral thesis (other academic/artistic)abstract
- The theme of this thesis is combinatorics, complex analysis and algebraic geometry. The thesis consists of six articles divided into four parts.Part A: Spectral properties of the Schrödinger equationThis part consists of Papers I-II, where we study a univariate Schrödinger equation with a complex polynomial potential. We prove that the set of polynomial potentials that admit solutions to the Schrödingerequation is connected, under certain boundary conditions. We also study a similar result for even polynomial potentials, where a similar result is obtained.Part B: Graph monomials and sums of squaresIn this part, consisting of Paper III, we study natural bases for the space of homogeneous, symmetric and translation-invariant polynomials in terms of multigraphs. We find all multigraphs with at most six edges that give rise to non-negative polynomials, and which of these that can be expressed as a sum of squares. Such polynomials appear naturally in connection to expressing certain non-negative polynomials as sums of squares.Part C: Eigenvalue asymptotics of banded Toeplitz matricesThis part consists of Papers IV-V. We give a new and generalized proof of a theorem by P. Schmidt and F. Spitzer concerning asymptotics of eigenvalues of Toeplitz matrices. We also generalize the notion of eigenvalues to rectangular matrices, and partially prove the a multivariate analogue of the above.Part D: Stretched Schur polynomials This part consists of Paper VI, where we give a combinatorial proof that certain sequences of skew Schur polynomials satisfy linear recurrences with polynomial coefficients.
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| 5. |
- Andersson, Ole
(author)
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Holonomy in Quantum Information Geometry
- 2018
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Licentiate thesis (other academic/artistic)abstract
- In this thesis we provide a uniform treatment of the two most popular non-adiabatic geometric phases for dynamical systems of mixed quantum states, namely those of Uhlmann and of Sjöqvist et al. We develop a holonomy theory for the latter which we also relate to the already existing theory for the former. This makes it clear what the similarities and differences between the two geometric phases are. We discuss and motivate constraints on the two phases. Furthermore, we discuss some topological properties of the holonomy of `real' quantum systems, and we introduce higher-order geometric phases for not necessarily cyclic dynamical systems of mixed states. In a final chapter we apply the theory developed for the geometric phase of Sjöqvist et al. to geometric uncertainty relations, including some new "quantum speed limits''.
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| 6. |
- Bergkvist, Tanja, 1974-
(author)
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Asymptotics of Eigenpolynomials of Exactly-Solvable Operators
- 2007
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Doctoral thesis (other academic/artistic)abstract
- The main topic of this doctoral thesis is asymptotic properties of zeros in polynomial families arising as eigenfunctions to exactly-solvable differential operators. The study was initially inspired by a number of striking results from computer experiments performed by G. Masson and B. Shapiro for a more restrictive class of operators. Our research is also motivated by a classical question going back to S. Bochner on a general classification of differential operators possessing an infinite sequence of orthogonal eigenpolynomials. In general however, the sequence of eigenpolynomials of an exactly-solvable operator is not an orthogonal system and it can therefore not be studied by means of the extensive theory known for such systems. Our study can thus be considered as the first steps to a natural generalization of the asymptotic behaviour of the roots of classical orthogonal polynomials. Exactly-solvable operators split into two major classes: non-degenerate and degenerate. We prove that in the former case, as the degree tends to infinity, the zeros of the eigenpolynomial are distributed according to a certain probability measure which is compactly supported on a tree and which depends only on the leading term of the operator. Computer experiments indicate the existence of a limiting root measure in the degenerate case too, but that it is compactly supported (conjecturally on a tree) only after an appropriate scaling which is conjectured (and partially proved) in this thesis. One of the main technical tools in this thesis is the Cauchy transform of a probability measure, which in the considered situation satisfies an algebraic equation. Due to the connection between the asymptotic root measure and its Cauchy transform it is therefore possible to obtain detailed information on the limiting zero distribution.
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| 7. |
- Forsgård, Jens, 1984-
(author)
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Tropical aspects of real polynomials and hypergeometric functions
- 2015
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Doctoral thesis (other academic/artistic)abstract
- The present thesis has three main topics: geometry of coamoebas, hypergeometric functions, and geometry of zeros.First, we study the coamoeba of a Laurent polynomial f in n complex variables. We define a simpler object, which we call the lopsided coamoeba, and associate to the lopsided coamoeba an order map. That is, we give a bijection between the set of connected components of the complement of the closed lopsided coamoeba and a finite set presented as the intersection of an affine lattice and a certain zonotope. Using the order map, we then study the topology of the coamoeba. In particular, we settle a conjecture of M. Passare concerning the number of connected components of the complement of the closed coamoeba in the case when the Newton polytope of f has at most n+2 vertices.In the second part we study hypergeometric functions in the sense of Gel'fand, Kapranov, and Zelevinsky. We define Euler-Mellin integrals, a family of Euler type hypergeometric integrals associated to a coamoeba. As opposed to previous studies of hypergeometric integrals, the explicit nature of Euler-Mellin integrals allows us to study in detail the dependence of A-hypergeometric functions on the homogeneity parameter of the A-hypergeometric system. Our main result is a complete description of this dependence in the case when A represents a toric projective curve.In the last chapter we turn to the theory of real univariate polynomials. The famous Descartes' rule of signs gives necessary conditions for a pair (p,n) of integers to represent the number of positive and negative roots of a real polynomial. We characterize which pairs fulfilling Descartes' conditions are realizable up to degree 7, and we provide restrictions valid in arbitrary degree.
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| 8. |
- Oneto, Alessandro, 1988-
(author)
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Waring-type problems for polynomials : Algebra meets Geometry
- 2016
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Doctoral thesis (other academic/artistic)abstract
- In the present thesis we analyze different types of additive decompositions of homogeneous polynomials. These problems are usually called Waring-type problems and their story go back to the mid-19th century and, recently, they received the attention of a large community of mathematicians and engineers due to several applications. At the same time, they are related to branches of Commutative Algebra and Algebraic Geometry.The classical Waring problem investigates decompositions of homogeneous polynomials as sums of powers of linear forms. Via Apolarity Theory, the study of these decompositions for a given polynomial F is related to the study of configuration of points apolar to F, namely, configurations of points whose defining ideal is contained in the ``perp'' ideal associated to F. In particular, we analyze which kind of minimal set of points can be apolar to some given polynomial in cases with small degrees and small number of variables. This let us introduce the concept of Waring loci of homogeneous polynomials.From a geometric point of view, questions about additive decompositions of polynomials can be described in terms of secant varieties of projective varieties. In particular, we are interested in the dimensions of such varieties. By using an old result due to Terracini, we can compute these dimensions by looking at the Hilbert series of homogeneous ideal.Hilbert series are very important algebraic invariants associated to homogeneous ideals. In the case of classical Waring problem, we have to look at power ideals, i.e., ideals generated by powers of linear forms. Via Apolarity Theory, their Hilbert series are related to Hilbert series of ideals of fat points, i.e., ideals of configurations of points with some multiplicity. In this thesis, we consider some special configuration of fat points. In general, Hilbert series of ideals of fat points is a very active field of research. We explain how it is related to the famous Fröberg's conjecture about Hilbert series of generic ideals.Moreover, we use Fröberg's conjecture to deduce the dimensions of several secant varieties of particular projective varieties and, then, to deduce results regarding some particular Waring-type problems for polynomials.In this thesis, we mostly work over the complex numbers. However, we also analyze the case of classical Waring decompositions for monomials over the real numbers. In particular, we classify for which monomials the minimal length of a decomposition in sum of powers of linear forms is independent from choosing the ground field as the field of complex or real numbers.
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| 9. |
- Alexandersson, Per, 1987-
(author)
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On eigenvalues of the Schrödinger operator with a complex-valued polynomial potential
- 2010
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Licentiate thesis (other academic/artistic)abstract
- In this thesis, we generalize a recent result of A. Eremenko and A. Gabrielov on irreducibility of the spectral discriminant for the Schroedinger equation with quartic potentials. In the first paper, we consider the eigenvalue problem with a complex-valued polynomial potential of arbitrary degree d and show that the spectral determinant of this problem is connected and irreducible. In other words, every eigenvalue can be reached from any other by analytic continuation. We also prove connectedness of the parameter spaces of the potentials that admit eigenfunctions satisfying k > 2 boundary conditions, except for the case d is even and k = d/2. In the latter case, connected components of the parameter space are distinguished by the number of zeros of the eigenfunctions. In the second paper, we only consider even polynomial potentials, and show that the spectral determinant for the eigenvalue problem consists of two irreducible components. A similar result to that of paper I is proved for k boundary conditions.
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| 10. |
- Nenashev, Gleb, 1992-
(author)
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Around power ideals : From Fröberg's conjecture to zonotopal algebra
- 2018
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Doctoral thesis (other academic/artistic)abstract
- In this thesis we study power algebras, which are quotient of polynomial rings by power ideals. We will study Hilbert series of such ideals and their other properties. We consider two important special cases, namely, zonotopal ideals and generic ideals. Such ideals have a lot combinatorial properties.In the first chapter we study zonotopal ideals, which were defined and used in several earlier publications. The most important works are by F.Ardila and A.Postnikov and by O.Holtz and A.Ron. These papers originate from different sources, the first source is homology theory, the second one is the theory of box splines. We study quotient algebras by these ideals; these algebras have a nice interpretation for their Hilbert series, as specializations of their Tutte polynomials. There are two important subclasses of these algebras, called unimodular and graphical. The graphical algebras were defined by A.Postnikov and B.Shapiro. In particular, the external algebra of a complete graph is exactly the algebra generated by the Bott-Chern forms of the corresponding complete flag variety. One of the main results of the thesis is a characterization of external algebras. In fact, for the case of graphical and unimodular algebras we prove that external algebras are in one-to-one correspondence with graphical and regular matroids, respectively.In the second chapter we study Hilbert series of generic ideals. By a generic ideal we mean an ideal generated by forms from some class, whose coefficients belong to a Zariski-open set. There are two main classes to consider: the first class is when we fix the degrees of generators; the famous Fröberg's conjecture gives the expected Hilbert series of such ideals; the second class is when an ideal is generated by powers of generic linear forms. There are a few partial results on Fröberg's conjecture, namely, when the number of variables is at most three. In both classes the Hilbert series is known in the case when the number of generators is at most (n+1). In both cases we construct a lot of examples when the degree of generators are the same and the Hilbert series is the expected one.
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