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  • 1.
    Ekedahl, Torsten
    et al.
    Stockholm Univeristy, dept of Mathematics.
    Laksov, Dan
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Splitting algebras, symmetric functions and Galois theory2005In: Journal of Algebra and its Applications, ISSN 0219-4988, E-ISSN 1793-6829, Vol. 4, no 1, p. 59-75Article in journal (Refereed)
    Abstract [en]

    We present a theory for splitting algebras of monic polynomials over rings, and apply the results to symmetric functions, and Galois theory.

  • 2.
    Gatto, L.
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.). Politecnico di Torino, Italy.
    Laksov, Dan
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.). Politecnico di Torino, Italy.
    From linear recurrence relations to linear ODEs with constant coefficients2016In: Journal of Algebra and its Applications, ISSN 0219-4988, E-ISSN 1793-6829, Vol. 15, no 6, article id 1650109Article in journal (Refereed)
    Abstract [en]

    Linear Ordinary Differential Equations (ODEs) with constant coefficients are studied by looking in general at linear recurrence relations in a module with coefficients in an arbitrary ℤalgebra. The bridge relating the two theories is the notion of formal Laplace transform associated to a sequence of invertibles. From this more economical perspective, generalized Wronskians associated to solutions of linear ODEs will be revisited, mentioning their relationships with Schubert Calculus for Grassmannians.

  • 3. Gustavsen, T. S.
    et al.
    Laksov, Dan
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Skjelnes, Roy Mikael
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    An elementary, explicit, proof of the existence of Hilbert schemes of points2007In: Journal of Pure and Applied Algebra, ISSN 0022-4049, E-ISSN 1873-1376, Vol. 210, no 3, p. 705-720Article in journal (Refereed)
    Abstract [en]

    There are many beautiful constructions of the Hilbert scheme of closed subschemes of projective schemes. In many cases where Hilbert schemes are involved, it suffices to know they exist. On the other hand, there are many situations when it is crucial to have an explicit description of the Hilbert schemes. In this article we give a simple construction of Hilbert schemes of points, under general circumstances, that provides such a description. In addition to being useful for computations the construction is short, elementary, and explicit. Our methods rely on simple algebraic constructions. It gives explicit expressions of members of an affine covering of the Hilbert schemes, involving few variables satisfying natural equations. In particular, it explains the connections with commuting schemes of matrices. We also give some examples showing how our method can be used.

  • 4. Gustavsen, Trond Stolen
    et al.
    Laksov, Dan
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Skjelnes, Roy Mikael
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    An elementary, explicit, proof of the existence of quot schemes of points2007In: Pacific Journal of Mathematics, ISSN 0030-8730, E-ISSN 1945-5844, Vol. 231, no 2, p. 401-415Article in journal (Refereed)
    Abstract [en]

    We give an easy and elementary construction of quotient schemes of modules of relatively finite rank under very general conditions. The construction provides a natural, explicit description of an affine covering of the quotient schemes, that is useful in many situation.

  • 5.
    Laksov, Dan
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    A formalism for equivariant Schubert calculus2009In: Algebra and Number Theory, ISSN 1937-0652, Vol. 3, no 6, p. 711-727Article in journal (Refereed)
    Abstract [en]

    In previous work we have developed a general formalism for Schubert calculus. Here we show how this theory can be adapted to give a formalism for equivariant Schubert calculus consisting of a basis theorem, a Pieri formula and a Giambelli formula. Our theory specializes to a formalism for equivariant cohomology of grassmannians. We interpret the results in a ring that can be considered as the formal generalized analog of localized equivariant cohomology of infinite grassmannians.

  • 6.
    Laksov, Dan
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    A Relation Between Symmetric Polynomials and the Algebra of Classes, Motivated by Equivariant Schubert Calculus2011In: Combinatorial Aspects of Commutative Algebra and Algebraic Geometry / [ed] Floystad, G; Johnsen, T; Knutsen, AL, Springer Berlin/Heidelberg, 2011, p. 89-97Conference paper (Refereed)
    Abstract [en]

    In previous work we developed a general formalism for equivariant Schubert calculus of grassmannians consisting of a basis theorem, a Pieri formula and a Giambelli formula. Part of the work consists in interpreting the results in a ring that can be considered as the formal generalized analog of localized equivariant cohomology of infinite grassmannians. Here we present an extract of the theory containing the essential features of this ring. In particular we emphasize the importance of the GKM condition. Our formalism and methods are influenced by the combinatorial formalism given by A. Knutson and T. Tao for equivariant cohomology of grassmannians, and of the use of factorial Schur polynomials in the work of L.C. Mihalcea.

  • 7.
    Laksov, Dan
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Diagonalization of matrices over rings2013In: Journal of Algebra, ISSN 0021-8693, E-ISSN 1090-266X, Vol. 376, p. 123-138Article in journal (Refereed)
    Abstract [en]

    We propose a method for diagonalizing matrices with entries, in commutative rings. The point of departure is to split the characteristic polynomial of the matrix over a (universal) splitting algebra, and to use the resulting universal roots to construct eigenvectors of the matrix. A crucial point is to determine when the determinant of the eigenvector matrix, that is the matrix whose columns are the eigenvectors, is regular in the splitting algebra. We show that this holds when the matrix is generic, that is, the entries are algebraically independent over the base ring. It would have been desirable to have an explicit formula for the determinant in the generic case. However, we have to settle for such a formula in a special case that is general enough for proving regularity in the general case. We illustrate the uses of our results by proving the Spectral Mapping Theorem, and by generalizing a fundamental result from classical invariant theory.

  • 8.
    Laksov, Dan
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Inverse systems, generic linear forms, divided powers, and transversality2013In: Journal of Pure and Applied Algebra, ISSN 0022-4049, E-ISSN 1873-1376, Vol. 217, no 12, p. 2255-2262Article in journal (Refereed)
    Abstract [en]

    The main result of this work is a transversality result for linear spaces of forms of polynomials vanishing on generic points. In the process of proving this result, we clarify and refine previous work with Froberg on the theory of inverse systems and generic forms. The needs for these improvements grew out of our proof of Zanello's bound for Hilbert functions.

  • 9.
    Laksov, Dan
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Linear maps and linear recursion2007In: Normat, ISSN 0801-3500, Vol. 55, no 2, p. 59-77Article in journal (Refereed)
  • 10.
    Laksov, Dan
    KTH, Superseded Departments, Mathematics.
    Local and global structure of effective and cuspidal loci on Grassmannians2003In: Communications in Algebra, ISSN 0092-7872, E-ISSN 1532-4125, Vol. 31, no 8, p. 3993-4006Article in journal (Refereed)
    Abstract [en]

    We introduce effective loci, which are central for the study of differential systems, in a very general setting. The greater generality makes them useful in a wider range of situations; and it also makes their study more straightforward, and convenient technically. Our treatment of effective loci is based on families of relative tangents. Such families give rise, in natural ways, to Semple sheaves, and hence allow us to construct the higher contact loci by straightforward iteration. Our main result is a local description of generic effective loci, on key Grassmannians for geometric applications. We show, by means of very natural frames, that these effective loci are precisely where certain generic matrices become symmetric.

  • 11.
    Laksov, Dan
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Norms on rings and the Hilbert scheme of points on the line2005In: Quarterly Journal of Mathematics, ISSN 0033-5606, E-ISSN 1464-3847, Vol. 56, p. 367-375Article in journal (Refereed)
    Abstract [en]

    We generalize the algebraic results of Laksov and Skjelnes (2001) and Skjelnes (2002), and obtain easy and transparent proofs of the representability of the Hilbert functor of points on the affine scheme whose coordinate ring is any localization of the polynomial ring in one variable over an arbitrary base ring. The coordinate ring of the Hilbert scheme is determined. We also make explicit the relation between our methods and the beautiful treatment of the Hilbert scheme of curves via norms, indicated by Grothendieck (1995), and performed by Deligne (1973).

  • 12.
    Laksov, Dan
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    On zanello's lower bound for level algebras2013In: Proceedings of the American Mathematical Society, ISSN 0002-9939, E-ISSN 1088-6826, Vol. 141, no 5, p. 1519-1527Article in journal (Refereed)
    Abstract [en]

    We consider the proof of Söderberg of Zanello's lower bound for the Hilbert function of level algebras from the point of view of vector spaces. Our results, when specialised to level algebras, generalise those of Zanello and Söderberg to the case when the modules involved may have nontrivial annihilators. In the process we clarify why the methods of Zanello and Söderberg consist of two distinct parts. As a contrast we show that for polynomial rings, Zanello's bound, in the generic case, can be obtained by simple manipulations of numbers without dividing into two separate cases. We also consider the inclusion-exclusion principle of dimensions of vector spaces used by Zanello in special cases. It turns out that the resulting alternating sums are extremely difficult to handle and have many unexpected properties. This we illustrate by a couple of results and examples. The examples show that the inclusion-exclusion principle does not hold for vector spaces in the way it is used by Zanello.

  • 13.
    Laksov, Dan
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Radicals of ideals that are not the intersection of radical primes2005In: Fundamenta Mathematicae, ISSN 0016-2736, E-ISSN 1730-6329, Vol. 185, no 1, p. 83-96Article in journal (Refereed)
    Abstract [en]

    Various kinds of radicals of ideals in commutative rings with identity appear in many parts of algebra and geometry, in particular in connection with the Hilbert Nullstellensatz, both in the noetherian and the non-noetherian case. All of these radicals, except the *-radicals, have the fundamental, and very useful, property that the radical of an ideal is the intersection of radical primes, that is, primes that are equal to their own radical. It is easy to verify that when the ring A is noetherian then the *-radical R(J) of an ideal is the intersection of *-radical primes. However, it has been an open question whether this holds in general. The main purpose of this article is to give an example of a ring with a *-radical that is not radical. To our knowledge it is the first example of a natural radical on a ring such that the radical of each ideal is not the intersection of radical primes. More generally, we present a method that may be used to construct more such examples. The main new idea is to introduce radical operations on the closed sets of topological spaces. We can then use the Zariski topology on the spectrum of a ring to translate algebraic questions into topology. It turns out that the quite intricate algebraic manipulations involved in handling the *-radical become much more transparent when rephrased in geometric terms.

  • 14.
    Laksov, Dan
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Schubert calculus and equivariant cohomology of grassmannians2008In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 217, no 4, p. 1869-1888Article in journal (Refereed)
    Abstract [en]

    We give a description of equivariant cohomology of grassmannians that places the theory into a general framework for cohomology theories of grassmannians. As a result we obtain a formalism for equivariant cohomology where the basic results of equivariant Schubert calculus, the basis theorem, Pieri's formula and Giambelli's formula can be obtained from the corresponding results of the general framework by a change of basis. In order to show that our formalism reflects the geometry of grassmannians we relate our theory to the treatment of equivariant cohomology of grassmannians by A. Knutson and T. Tao.

  • 15.
    Laksov, Dan
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Splitting algebras and Gysin homomorphisms2010In: Journal of Commutative Algebra, ISSN 1939-0807, E-ISSN 1939-2346, Vol. 2, no 3, p. 401-425Article in journal (Refereed)
    Abstract [en]

    We give three algebraic constructions of splitting algebras of monic polynomials with coefficients in arbitrary commutative rings with a unit, and of the corresponding Gysin homomorphisms.

  • 16.
    Laksov, Dan
    KTH, Superseded Departments, Mathematics.
    The algebra of jets2000In: The Michigan mathematical journal, ISSN 0026-2285, E-ISSN 1945-2365, Vol. 48, p. 393-416Article in journal (Refereed)
  • 17.
    Laksov, Dan
    KTH, Superseded Departments, Mathematics.
    Wronski systems for families of local complete intersection curves2003In: Communications in Algebra, ISSN 0092-7872, E-ISSN 1532-4125, Vol. 31, no 8, p. 4007-4035Article in journal (Refereed)
    Abstract [en]

    We construct a substitute for principal parts for systems of algebras with a natural local presentation by free modules. The main application of the theory is to families of locally complete intersection curves. In that case we obtain the sheaves of E steves. The advantage of the greater generality is that it makes the theory much more transparent. It also makes it possible to avoid many technical difficulties.

  • 18.
    Laksov, Dan
    et al.
    KTH, Superseded Departments, Mathematics.
    Ekedahl, Torsten
    Two "generic" proofs of the spectral mapping theorem2004In: The American mathematical monthly, ISSN 0002-9890, E-ISSN 1930-0972, Vol. 111, no 7, p. 572-585Article in journal (Refereed)
  • 19.
    Laksov, Dan
    et al.
    KTH, Superseded Departments, Mathematics.
    Pitteloud, Y.
    Skjelnes, Roy Mikael
    KTH, Superseded Departments, Mathematics.
    Notes on flatness and the Quot functor on rings2000In: Communications in Algebra, ISSN 0092-7872, E-ISSN 1532-4125, Vol. 28, no 12, p. 5613-5627Article in journal (Refereed)
    Abstract [en]

    For Rat modules M over a ring A we study the similarities between the three statements, dim(kappa (P))(kappa (P) circle times (A) M) = d far all prime ideals P of A, the A(P)-module Mp is free of rank d for all prime ideals P of A, and M is a locally free A-module of rank d. We have particularly emphasized the case when there is an A-algebra B, essentially of finite type, and M is a finitely generated B-module.

  • 20.
    Laksov, Dan
    et al.
    KTH, Superseded Departments, Mathematics.
    Skjelnes, Roy Mikael
    KTH, Superseded Departments, Mathematics.
    The Hilbert scheme parameterizing finite length subschemes of the line with support at the origin2001In: Compositio Mathematica, ISSN 0010-437X, E-ISSN 1570-5846, Vol. 126, no 3, p. 323-334Article in journal (Refereed)
    Abstract [en]

    We introduce symmetrizing operators of the polynomial ring A[x] in the variable x over a ring A. When A is an algebra over a field k these operators are used to characterize the monic polynomials F(x) of degree n in A[x] such that Ax(k)k[x]((x))/(F(x)) is a free A-module of rank n. We use the characterization to determine the Hilbert scheme parameterizing subschemes of length n of k[x]((x)).

  • 21.
    Laksov, Dan
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Svensson, Lars
    Thorup, Anders
    Copenhagen University, Dept. of Mathematics.
    The spectral mapping theorem, norms on rings, and resultants2000In: L'Enseignement Mathématique. Revue Internationale. 2e Série, ISSN 0013-8584, Vol. 46, no 3-4, p. 349-358Article in journal (Refereed)
  • 22.
    Laksov, Dan
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Thorup, Anders
    A determinantal formula for the exterior powers of the polynomial ring2007In: Indiana University Mathematics Journal, ISSN 0022-2518, E-ISSN 1943-5258, Vol. 56, no 2, p. 825-845Article in journal (Refereed)
    Abstract [en]

    We present a very general, conceptually natural, explicit and computationally efficient Schubert calculus. It consists of two strongly interrelated parts, a structure theorem for an exterior power of the polynomial ring in one variable as a module over the ring of symmetric polynomials and a determinantal formula. Both parts are similar to corresponding results in algebra, combinatorics and geometry. To emphasize the connections with these fields we give several proofs of our main results, each proof illuminating the theory from a different algebraic, combinatorial, or geometric angle. The main application of our theory is to the Schubert calculus of Grassmann schemes, where it gives a natural homology and cohomology theory for the Grassmannians in a very general setting.

  • 23.
    Laksov, Dan
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Thorup, Anders
    Schubert Calculus on Grassmannians and Exterior Powers2009In: Indiana University Mathematics Journal, ISSN 0022-2518, E-ISSN 1943-5258, Vol. 58, no 1, p. 283-300Article in journal (Refereed)
    Abstract [en]

    We present a general algebraic formalism for homology and cohomology theories for Grassmannians. The formalism is expressed in terms of the action of factorization algebras on exterior powers.

  • 24.
    Laksov, Dan
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Thorup, Anders
    Splitting Algebras and Schubert Calculus2012In: Indiana University Mathematics Journal, ISSN 0022-2518, E-ISSN 1943-5258, Vol. 61, no 3, p. 1253-1312Article in journal (Refereed)
    Abstract [en]

    We continue previous work on Schubert calculus of Grassmann schemes via splitting and factorization algebras. The aim of the project is to describe the intersection theory of flag schemes under very general conditions. In this part we extend, generalize, and refine the previous work. Among the main accomplishments of this article is the introduction of Schur determinants generalizing Schur polynomials. For the Schur determinants we obtain determinantal formulas, polarity formulas, and Gysin formulas in great generality. Our main tools are, as in our previous works, splitting and factorization algebras, residues of finite sets of Laurent series, and Gysin maps. The tools provide flexible techniques, useful in many parts of algebra, combinatorics, geometry and representation theory.

1 - 24 of 24
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