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University of Notre Dame

1. Bernadette M. Boyle. On the Unimodality of Pure O-Sequences</h1>.

Degree: Mathematics, 2012, University of Notre Dame

URL: https://curate.nd.edu/show/n296ww74n34

In this dissertation, we will discuss some
properties of pure O-sequences, which, due to Macaulay’s Inverse
Systems, are in bijective correspondence with the Hilbert functions
of Artinian level monomial algebras. In particular, we will focus
on their unimodality. A sequence of numbers is unimodal if it does
not increase after a strict decrease. There has
been previous progress in this area. Specifically, it is known that
all algebras in two variables are unimodal, due to Macaulay’s
Maximal Growth Theorem. Furthermore, it is known that all pure
O-sequences of type two in three variables are unimodal and the
Hilbert function of monomial complete intersections are unimodal;
these results are due to the Weak Lefschetz Property. In addition
to these families of pure O-sequences which are unimodal, there are
known families of pure O-sequences which fail to be unimodal. In
particular, for any r > 2, there exists a monomial Artinian
level algebra in r variables whose Hilbert function fails
unimodality with an arbitrary number of peaks.
We will focus on the question of whether particular socle
types or socle degrees in a fixed number of variables guarantee
unimodality. Since the Weak Lefschetz Property is only guaranteed
to hold in the cases mentioned above, we will use different
techniques in our approach. We will show that all pure O-sequences
of type three in three variables and all pure O-sequence of type
two in four variables are strictly unimodal. We will also show that
all pure O-sequences with socle degree less than or equal to nine
in three variables are unimodal and all pure O-sequences (except
for possibly two cases) with socle degree less than or equal to
four in four variables are unimodal. Finally we will show that for
r greater than or equal to 4 and e greater than or equal to 7,
there exists a non-unimodal pure O-sequence in r variables with
socle degree e.
*Advisors/Committee Members: Dr. Claudia Polini, Committee Member, Dr. Juan Migliore, Committee Chair, Dr. Nero Budur, Committee Member, Dr. Yu Xie, Committee Member.*

Subjects/Keywords: pure O-sequence; Hilbert function; unimodal; level monomial Artinian algebra

Record Details Similar Records

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

Boyle, B. M. (2012). On the Unimodality of Pure O-Sequences</h1>. (Thesis). University of Notre Dame. Retrieved from https://curate.nd.edu/show/n296ww74n34

Note: this citation may be lacking information needed for this citation format:

Not specified: Masters Thesis or Doctoral Dissertation

Chicago Manual of Style (16^{th} Edition):

Boyle, Bernadette M.. “On the Unimodality of Pure O-Sequences</h1>.” 2012. Thesis, University of Notre Dame. Accessed July 07, 2020. https://curate.nd.edu/show/n296ww74n34.

Note: this citation may be lacking information needed for this citation format:

Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7^{th} Edition):

Boyle, Bernadette M.. “On the Unimodality of Pure O-Sequences</h1>.” 2012. Web. 07 Jul 2020.

Vancouver:

Boyle BM. On the Unimodality of Pure O-Sequences</h1>. [Internet] [Thesis]. University of Notre Dame; 2012. [cited 2020 Jul 07]. Available from: https://curate.nd.edu/show/n296ww74n34.

Note: this citation may be lacking information needed for this citation format:

Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Boyle BM. On the Unimodality of Pure O-Sequences</h1>. [Thesis]. University of Notre Dame; 2012. Available from: https://curate.nd.edu/show/n296ww74n34

Not specified: Masters Thesis or Doctoral Dissertation

University of Notre Dame

2. Bonnie Bradberry Smith. Cores of Monomial Ideals</h1>.

Degree: Mathematics, 2010, University of Notre Dame

URL: https://curate.nd.edu/show/mg74qj74z7p

In this dissertation, we describe the cores
of several classes of monomial ideals. We also find bounds on the
reduction numbers of these ideals. The first class of ideals which
we consider is one coming from graph theory, the strongly stable
ideals of degree two. We prove that a strongly stable ideal I of
degree two in k[x*1,…,x*d] has the G*d
property if and only if x*{g-1}x*d is in I,
where g is the height of I. We also prove that any strongly stable
ideal of degree two in k[x*1,…,x*d] which has
the G*d property satisfies the Artin-Nagata property
AN*{d-1}. We then show that the core of such an ideal I is
given by core(I)=I m*^{g-1}, where g is the height of I, and m is
the homogeneous maximal ideal (x1,…,x*d). We
also show that the reduction number r(I) is at most g-1.
Specifically, we prove that an ideal J=J*I whose elements
correspond to diagonals in the tableau associated to I is a minimal
reduction of I, and that r_{J}(I) < g.
The other classes of ideals which we
consider are zero-dimensional monomial ideals in the polynomial
ring R=k[x*1,…,x*d]. We show that, if such an
ideal I is an almost complete intersection, then the core of I
satisfies a (d+1)-fold symmetry property coming from the generators
of I, and hence that the shape of core(I) is closely related to the
shape of I. Using this symmetry property, we completely describe
the shape of core(I) in the case where I has a monomial minimal
reduction. Then, in the two-dimensional case, we give an algorithm
for computing the core of I which allows us to prove that the
minimal number of generators of core(I) is 2r+2, where r is the
reduction number of I. We prove that this result holds even for
ideals which do not have a minimal reduction which is monomial. We
also describe the core of an ideal I in k[x,y] having more
generators, in the case where I has a monomial minimal reduction.
Finally, we consider a strongly stable ideal I in k[x,y] having a
monomial minimal reduction J. We prove that r_{J}(I) leq mu(I)-2, and
we give an algorithm for obtaining the core of I via its first
coefficient ideal.
*Advisors/Committee Members: Claudia Polini, Committee Chair, Yu Xie, Committee Member, Nero Budur, Committee Member, Juan Migliore, Committee Member.*

Subjects/Keywords: strongly stable ideals; almost complete intersections; cores of ideals; reductions of ideals

Record Details Similar Records

❌

APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

Smith, B. B. (2010). Cores of Monomial Ideals</h1>. (Thesis). University of Notre Dame. Retrieved from https://curate.nd.edu/show/mg74qj74z7p

Not specified: Masters Thesis or Doctoral Dissertation

Chicago Manual of Style (16^{th} Edition):

Smith, Bonnie Bradberry. “Cores of Monomial Ideals</h1>.” 2010. Thesis, University of Notre Dame. Accessed July 07, 2020. https://curate.nd.edu/show/mg74qj74z7p.

Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7^{th} Edition):

Smith, Bonnie Bradberry. “Cores of Monomial Ideals</h1>.” 2010. Web. 07 Jul 2020.

Vancouver:

Smith BB. Cores of Monomial Ideals</h1>. [Internet] [Thesis]. University of Notre Dame; 2010. [cited 2020 Jul 07]. Available from: https://curate.nd.edu/show/mg74qj74z7p.

Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Smith BB. Cores of Monomial Ideals</h1>. [Thesis]. University of Notre Dame; 2010. Available from: https://curate.nd.edu/show/mg74qj74z7p

Not specified: Masters Thesis or Doctoral Dissertation

University of Notre Dame

3. Angela Kohlhaas. The core of an ideal and its relationship to the adjoint and coefficient ideals</h1>.

Degree: Mathematics, 2010, University of Notre Dame

URL: https://curate.nd.edu/show/bv73bz62h3d

Given an ideal I in a Noetherian ring R,
the core of I is the intersection of all ideals contained in I
with the same integral closure as I. The core naturally arises in
the context of the Brianc{c}on-Skoda theorem as an ideal which
contains the adjoint of a certain power of I. As the
arbitrary-characteristic analog of the multiplier ideal, the
adjoint is an important tool in the study of resolutions of
singularities. The question of when the core and the adjoint of a
power of I are equal has been tied to a celebrated conjecture of
Kawamata about the non-vanishing of sections of line bundles. We
show for certain classes of monomial ideals in the polynomial ring
k[x*1,ldots,x*d] over a field of
characteristic zero , core(I)=adj(I^{d}) if and only if core(I)
is integrally closed. In order to prove our main result, we further
develop the theory of coefficient ideals in regular local rings of
dimension two and study the combinatorial properties of the core of
a monomial ideal via the symmetry of its exponent
set.
*Advisors/Committee Members: Claudia Polini, Committee Member, Nero Budur, Committee Member, J. Chris Howk, Committee Chair, Juan Migliore, Committee Member, Yu Xie, Committee Member.*

Subjects/Keywords: exponent set; commutative algebra; birational geometry

Record Details Similar Records

❌

APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

Kohlhaas, A. (2010). The core of an ideal and its relationship to the adjoint and coefficient ideals</h1>. (Thesis). University of Notre Dame. Retrieved from https://curate.nd.edu/show/bv73bz62h3d

Not specified: Masters Thesis or Doctoral Dissertation

Chicago Manual of Style (16^{th} Edition):

Kohlhaas, Angela. “The core of an ideal and its relationship to the adjoint and coefficient ideals</h1>.” 2010. Thesis, University of Notre Dame. Accessed July 07, 2020. https://curate.nd.edu/show/bv73bz62h3d.

Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7^{th} Edition):

Kohlhaas, Angela. “The core of an ideal and its relationship to the adjoint and coefficient ideals</h1>.” 2010. Web. 07 Jul 2020.

Vancouver:

Kohlhaas A. The core of an ideal and its relationship to the adjoint and coefficient ideals</h1>. [Internet] [Thesis]. University of Notre Dame; 2010. [cited 2020 Jul 07]. Available from: https://curate.nd.edu/show/bv73bz62h3d.

Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Kohlhaas A. The core of an ideal and its relationship to the adjoint and coefficient ideals</h1>. [Thesis]. University of Notre Dame; 2010. Available from: https://curate.nd.edu/show/bv73bz62h3d

Not specified: Masters Thesis or Doctoral Dissertation