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  • 1. Altafi, Nasrin
    et al.
    Lundqvist, Samuel
    Stockholm University, Faculty of Science, Department of Mathematics.
    Monomial ideals and the failure of the Strong Lefschetz property2022In: Collectanea Mathematica (Universitat de Barcelona), ISSN 0010-0757, E-ISSN 2038-4815, Vol. 73, no 3, p. 383-390Article in journal (Refereed)
    Abstract [en]

    We give a sharp lower bound for the Hilbert function in degree d of artinian quotients k[x1,…,xn]/I failing the Strong Lefschetz property, where I is a monomial ideal generated in degree d≥2. We also provide sharp lower bounds for other classes of ideals, and connect our result to the classification of the Hilbert functions forcing the Strong Lefschetz property by Zanello and Zylinski.

  • 2. Boij, Mats
    et al.
    Fröberg, Ralf
    Stockholm University, Faculty of Science, Department of Mathematics.
    Lundqvist, Samuel
    Stockholm University, Faculty of Science, Department of Mathematics.
    Powers of generic ideals and the weak Lefschetz property for powers of some monomial complet intersections2018In: Journal of Algebra, ISSN 0021-8693, E-ISSN 1090-266X, Vol. 495, p. 1-14Article in journal (Refereed)
    Abstract [en]

    Given an ideal I = (f(1) ... , f(r)) in C[x(1), ... , x(n),] generated by forms of degree d, and an integer k > 1, how large can the ideal I-k be, i.e., how small can the Hilbert function of C[x(1), ... , x(n)] / I-k be? If r <= n the smallest Hilbert function is achieved by any complete intersection, but for r > n, the question is in general very hard to answer. We study the problem for r = n + 1, where the result is known for k = 1. We also study a closely related problem, the Weak Lefschetz property, for S/I-k, where I is the ideal generated by the d'th powers of the variables.

  • 3. Boij, Mats
    et al.
    Lundqvist, Samuel
    Stockholm University, Faculty of Science, Department of Mathematics.
    A classification of the weak Lefschetz property for almost complete intersections generated by uniform powers of general linear forms2023In: Algebra & Number Theory, ISSN 1937-0652, E-ISSN 1944-7833, Vol. 17, no 1, p. 111-126Article in journal (Refereed)
    Abstract [en]

    We use Macaulay’s inverse system to study the Hilbert series for almost complete intersections generated by uniform powers of general linear forms. This allows us to give a classification of the weak Lefschetz property for these algebras, settling a conjecture by Migliore, Miró-Roig, and Nagel.

  • 4. Ceria, Michela
    et al.
    Lundqvist, Samuel
    Stockholm University, Faculty of Science, Department of Mathematics.
    Mora, Teo
    Degröbnerization: a political manifesto2022In: Applicable Algebra in Engineering, Communication and Computing, ISSN 0938-1279, E-ISSN 1432-0622, Vol. 33, no 6, p. 675-723Article in journal (Refereed)
    Abstract [en]

    Computer Algebra relies heavily on the computation of Gröbner bases, and these computations are primarily performed by means of Buchberger’s algorithm. In this overview paper, we focus on methods avoiding the computational intensity associated to Buchberger’s algorithm and, in most cases, even avoiding the concept of Gröbner bases, in favour of methods relying on linear algebra and combinatorics.

  • 5. Crispin Quiñonez, Veronica
    et al.
    Lundqvist, Samuel
    Stockholm University, Faculty of Science, Department of Mathematics.
    Nenashev, Gleb
    On ideals generated by two generic quadratic forms in the exterior algebra2019In: Journal of Pure and Applied Algebra, ISSN 0022-4049, E-ISSN 1873-1376, Vol. 223, no 12, p. 5067-5082Article in journal (Refereed)
    Abstract [en]

    Based on the structure theory of pairs of skew-symmetric matrices, we give a conjecture for the Hilbert series of the exterior algebra modulo the ideal generated by two generic quadratic forms. We show that the conjectured series is an upper bound in the coefficient-wise sense, and we determine a majority of the coefficients. We also conjecture that the series is equal to the series of the squarefree polynomial ring modulo the ideal generated by the squares of two generic linear forms.

  • 6. Crispin Quiñonez, Veronica
    et al.
    Lundqvist, Samuel
    Stockholm University, Faculty of Science, Department of Mathematics.
    Nenashev, Gleb
    Stockholm University, Faculty of Science, Department of Mathematics.
    On ideals generated by two generic quadratic forms in the exterior algebraManuscript (preprint) (Other academic)
    Abstract [en]

    Based on the structure theory of pairs of skew-symmetric matrices, we give a conjecture for the Hilbert series of the exterior algebra modulo the ideal generated by two generic quadratic forms. We show that the conjectured series is an upper bound in the coefficient-wise sense, and we determine a majority of the coefficients. We also conjecture that the series is equal to the series of the squarefree polynomial ring modulo the ideal generated by the squares of two generic linear forms.

  • 7.
    Fröberg, Ralf
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Lundqvist, Samuel
    Stockholm University, Faculty of Science, Department of Mathematics.
    Questions and conjectures on extremal Hilbert series2018In: Revista de la Unión Matemática Argentina, ISSN 0041-6932, E-ISSN 1669-9637, Vol. 59, no 2, p. 415-429Article in journal (Refereed)
    Abstract [en]

    Given an ideal of forms in an algebra (polynomial ring, tensor algebra, exterior algebra, Lie algebra, bigraded polynomial ring), we consider the Hilbert series of the factor ring. We concentrate on the minimal Hilbertseries, which is achieved when the forms are generic. In the polynomial ring we also consider the opposite case of maximal series. This is mainly a survey article, but we give a lot of problems and conjectures. The only novel results concern the maximal series in the polynomial ring.

  • 8.
    Fröberg, Ralf
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Lundqvist, Samuel
    Stockholm University, Faculty of Science, Department of Mathematics.
    Oneto, Alessandro
    Shapiro, Boris
    Stockholm University, Faculty of Science, Department of Mathematics.
    Algebraic Stories from One and from the Other Pockets2018In: Arnold Mathematical Journal, ISSN 2199-6792, Vol. 4, no 2, p. 137-160Article in journal (Refereed)
    Abstract [en]

    In what follows, we present a large number of questions which were posed on the problem solving seminar in algebra at Stockholm University during the period Fall 2014—Spring 2017 along with a number of results related to these problems. Many of the results were obtained by participants of the latter seminar.

  • 9. Gasanova, Oleksandra
    et al.
    Lundqvist, Samuel
    Stockholm University, Faculty of Science, Department of Mathematics.
    Nicklasson, Lisa
    Stockholm University, Faculty of Science, Department of Mathematics. Università degli Studi di Genova, Italy.
    On decomposing monomial algebras with the Lefschetz properties2022In: Journal of Pure and Applied Algebra, ISSN 0022-4049, E-ISSN 1873-1376, Vol. 226, no 6, article id 106968Article in journal (Refereed)
    Abstract [en]

    We introduce a general technique for decomposing monomial algebras which we use to study the Lefschetz properties. In particular, we prove that Gorenstein codimension three algebras arising from numerical semigroups have the strong Lefschetz property, and we give partial results on monomial almost complete intersections. We also study the reverse of the decomposition process – a gluing operation – which gives a way to construct monomial algebras with the Lefschetz properties.

  • 10.
    Lundqvist, Samuel
    Stockholm University, Faculty of Science, Department of Mathematics.
    A Buchberger like algorithm without monomial orderings: the graded commutative caseManuscript (preprint) (Other academic)
    Abstract [en]

    We analyze an algorithm to compute vector space bases for graded commutative algebras. The algorithm does not require a monomial ordering, but we show that when we use a monomial ordering, there is a strong connection to the Buchberger algorithm.

  • 11.
    Lundqvist, Samuel
    Stockholm University, Faculty of Science, Department of Mathematics.
    An algorithm to determine the Hilbert series for graded associative algebrasManuscript (preprint) (Other academic)
    Abstract [en]

    In this paper we present an algorithm to compute the Hilbert series for quotients of the free algebra with homogeneous two-sided ideals. We also give a modified version of the algorithm for quotients of the polynomial ring. The algorithms are implemented in a computer program named ``aalg'' and we compare running times with other computer algebra programs.

  • 12.
    Lundqvist, Samuel
    Stockholm University, Faculty of Science, Department of Mathematics.
    Boolean ideals and their varieties2015In: Journal of Pure and Applied Algebra, ISSN 0022-4049, E-ISSN 1873-1376, Vol. 219, no 10, p. 4521-4540Article in journal (Refereed)
    Abstract [en]

    We consider ideals in the ring Z2[x1,…,xn] that contain the polynomials  for i=1,…,n and give various results related to the one-to-one correspondence between these ideals and the subsets of . We also study the standard monomials with respect to the lexicographical ordering for these ideals and derive a distribution result.

  • 13.
    Lundqvist, Samuel
    Stockholm University, Faculty of Science, Department of Mathematics.
    Complexity of comparing monomials and two improvements of the Buchberger-Möller algorithm2008In: Lecture Notes in Computer Science, ISSN 0302-9743, E-ISSN 1611-3349, Vol. 5393, p. 105-125Article in journal (Refereed)
    Abstract [en]

    We give a new algorithm for merging sorted lists of monomials. Together with a projection technique we obtain a new complexity bound for the Buchberger-Möller algorithm.

  • 14.
    Lundqvist, Samuel
    Stockholm University, Faculty of Science, Department of Mathematics.
    Computational algorithms for algebras2009Doctoral thesis, comprehensive summary (Other academic)
    Abstract [en]

    This thesis consists of six papers.

    In Paper I, we give an algorithm for merging sorted lists of monomials and together with a projection technique, we obtain a new complexity bound for the Buchberger-Möller algorithm and the FGLM algorithm.

    In Paper II, we discuss four different constructions of vector space bases associated to vanishing ideals of points. We show how to compute normal forms with respect to these bases and give complexity bounds. As an application we drastically improve the computational algebra approach to the reverse engineering of gene regulatory networks.

    In Paper III, we introduce the concept of multiplication matrices for ideals of projective dimension zero. We discuss various applications and, in particular, we give a new algorithm to compute the variety of an ideal of projective dimension zero.

    In Paper IV, we consider a subset of projective space over a finite field and give a geometric description of the minimal degree of a non-vanishing form with respect to this subset. We also give bounds on the minimal degree in terms of the cardinality of the subset.

    In Paper V, we study an associative version of an algorithm constructed to compute the Hilbert series for graded Lie algebras. In the commutative case we use Gotzmann's persistence theorem to show that the algorithm terminates in finite time.

    In Paper VI, we connect the commutative version of the algorithm in Paper V with the Buchberger algorithm.

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  • 15.
    Lundqvist, Samuel
    Stockholm University, Faculty of Science, Department of Mathematics.
    Multiplication matrices and ideals of projective dimension zero2012In: Mathematics in Computer Science, ISSN 1661-8270, Vol. 6, no 1, p. 43-59Article in journal (Refereed)
    Abstract [en]

    We introduce the concept of multiplication matrices for ideals of projective dimension zero. We discuss various applications and, in particular, we give a new algorithm to compute the variety of an ideal of projective dimension zero.

  • 16.
    Lundqvist, Samuel
    Stockholm University, Faculty of Science, Department of Mathematics.
    Non-vanishing forms in projective space over finite fields2010In: Journal of Commutative Algebra, ISSN 1939-2346, Vol. 2, no 4, p. 435-443Article in journal (Refereed)
    Abstract [en]

    We consider a subset of projective space over a finite field and give a geometric description of the minimal degree of a non-vanishing form with respect to this subset. We also give bounds on the minimal degree in terms of the cardinality of the subset.

  • 17.
    Lundqvist, Samuel
    Stockholm University, Faculty of Science, Department of Mathematics.
    Vector space bases associated to vanishing ideals of points2010In: Journal of Pure and Applied Algebra, ISSN 0022-4049, E-ISSN 1873-1376, Vol. 214, no 4, p. 309-321Article in journal (Refereed)
    Abstract [en]

    In this paper we discuss four different constructions of vector space bases associated to vanishing ideals of points. We show how to compute normal forms with respect to these bases and give new complexity bounds. As an application, we drastically improve the computational algebra approach to the reverse engineering of gene regulatory networks.

  • 18.
    Lundqvist, Samuel
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Nicklasson, Lisa
    Stockholm University, Faculty of Science, Department of Mathematics.
    On generic principal ideals in the exterior algebra2019In: Journal of Pure and Applied Algebra, ISSN 0022-4049, E-ISSN 1873-1376, Vol. 223, no 6, p. 2615-2634Article in journal (Refereed)
    Abstract [en]

    We give a lower bound on the Hilbert series of the exterior algebra modulo a principal ideal generated by a generic form of odd degree and disprove a conjecture by Moreno-Socias and Snellman. We also show that the lower bound is equal to the minimal Hilbert series in some specific cases.

  • 19.
    Lundqvist, Samuel
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Nicklasson, Lisa
    Stockholm University, Faculty of Science, Department of Mathematics.
    On the structure of monomial complete intersections in positive characteristic2019In: Journal of Algebra, ISSN 0021-8693, E-ISSN 1090-266X, Vol. 521, p. 213-234Article in journal (Refereed)
    Abstract [en]

    In this paper we study the Lefschetz properties of monomial complete intersections in positive characteristic. We give a complete classification of the strong Lefschetz property when the number of variables is at least three, which proves a conjecture by Cook II. We also extend earlier results on the weak Lefschetz property by dropping the assumption on the residue field being infinite, and by giving new sufficient criteria.

  • 20.
    Lundqvist, Samuel
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Oneto, Alessandro
    Reznick, Bruce
    Shapiro, Boris
    Stockholm University, Faculty of Science, Department of Mathematics.
    On generic and maximal k-ranks of binary forms2019In: Journal of Pure and Applied Algebra, ISSN 0022-4049, E-ISSN 1873-1376, Vol. 223, no 5, p. 2062-2079Article in journal (Refereed)
    Abstract [en]

    In what follows, we pose two general conjectures about decompositions of homogeneous polynomials as sums of powers. The first one (suggested by G. Ottaviani) deals with the generic k-rank of complex-valued forms of any degree divisible by k in any number of variables. The second one (by the fourth author) deals with the maximal k-rank of binary forms. We settle the first conjecture in the cases of two variables and the second in the first non-trivial case of the 3-rd powers of quadratic binary forms.

  • 21.
    Löfwall, Clas
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Lundqvist, Samuel
    Stockholm University, Faculty of Science, Department of Mathematics.
    Roos, Jan-Erik
    Stockholm University, Faculty of Science, Department of Mathematics.
    A Gorenstein numerical semi-group ring having a transcendental series of Betti numbers2015In: Journal of Pure and Applied Algebra, ISSN 0022-4049, E-ISSN 1873-1376, Vol. 219, no 3, p. 591-621Article in journal (Refereed)
    Abstract [en]

    We prove in particular that the Gorenstein numerical semigroup ring generated by (36,48,50,52,56,60,66,67,107,121,129,135) has a transcendental series of Betti numbers. The methods of proofs are the theory of Golod homomorphism,large maps and the theory of infinite positively graded Liealgebras.

  • 22.
    Löfwall, Clas
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Lundqvist, Samuel
    Stockholm University, Faculty of Science, Department of Mathematics.
    Roos, Jan-Erik
    Stockholm University, Faculty of Science, Department of Mathematics.
    A Gorenstein numerical semi-group ring having a transcendental series of Betti numbers2012Report (Other academic)
    Abstract [en]

    We prove in particular that the Gorenstein numerical semigroup ring generated by (36,48,50,52,56,60,66,67,107,121,129,135) has a transcendental series of Betti numbers. The methods of proofs are the theory of Golod homomorphisms and the theory of infinite positively graded Lie algebras.

1 - 22 of 22
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