---
_id: '2006'
abstract:
- lang: eng
text: 'The monotone secant conjecture posits a rich class of polynomial systems,
all of whose solutions are real. These systems come from the Schubert calculus
on flag manifolds, and the monotone secant conjecture is a compelling generalization
of the Shapiro conjecture for Grassmannians (Theorem of Mukhin, Tarasov, and Varchenko).
We present some theoretical evidence for this conjecture, as well as computational
evidence obtained by 1.9 teraHertz-years of computing, and we discuss some of
the phenomena we observed in our data. '
article_processing_charge: No
author:
- first_name: Nicolas
full_name: Hein, Nicolas
last_name: Hein
- first_name: Christopher
full_name: Hillar, Christopher
last_name: Hillar
- first_name: Abraham
full_name: Martin Del Campo Sanchez, Abraham
id: 4CF47F6A-F248-11E8-B48F-1D18A9856A87
last_name: Martin Del Campo Sanchez
- first_name: Frank
full_name: Sottile, Frank
last_name: Sottile
- first_name: Zach
full_name: Teitler, Zach
last_name: Teitler
citation:
ama: Hein N, Hillar C, Martin del Campo Sanchez A, Sottile F, Teitler Z. The monotone
secant conjecture in the real Schubert calculus. Experimental Mathematics.
2015;24(3):261-269. doi:10.1080/10586458.2014.980044
apa: Hein, N., Hillar, C., Martin del Campo Sanchez, A., Sottile, F., & Teitler,
Z. (2015). The monotone secant conjecture in the real Schubert calculus. Experimental
Mathematics. Taylor & Francis. https://doi.org/10.1080/10586458.2014.980044
chicago: Hein, Nicolas, Christopher Hillar, Abraham Martin del Campo Sanchez, Frank
Sottile, and Zach Teitler. “The Monotone Secant Conjecture in the Real Schubert
Calculus.” Experimental Mathematics. Taylor & Francis, 2015. https://doi.org/10.1080/10586458.2014.980044.
ieee: N. Hein, C. Hillar, A. Martin del Campo Sanchez, F. Sottile, and Z. Teitler,
“The monotone secant conjecture in the real Schubert calculus,” Experimental
Mathematics, vol. 24, no. 3. Taylor & Francis, pp. 261–269, 2015.
ista: Hein N, Hillar C, Martin del Campo Sanchez A, Sottile F, Teitler Z. 2015.
The monotone secant conjecture in the real Schubert calculus. Experimental Mathematics.
24(3), 261–269.
mla: Hein, Nicolas, et al. “The Monotone Secant Conjecture in the Real Schubert
Calculus.” Experimental Mathematics, vol. 24, no. 3, Taylor & Francis,
2015, pp. 261–69, doi:10.1080/10586458.2014.980044.
short: N. Hein, C. Hillar, A. Martin del Campo Sanchez, F. Sottile, Z. Teitler,
Experimental Mathematics 24 (2015) 261–269.
date_created: 2018-12-11T11:55:10Z
date_published: 2015-06-23T00:00:00Z
date_updated: 2021-01-12T06:54:40Z
day: '23'
department:
- _id: CaUh
doi: 10.1080/10586458.2014.980044
intvolume: ' 24'
issue: '3'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: http://arxiv.org/abs/1109.3436
month: '06'
oa: 1
oa_version: Preprint
page: 261 - 269
publication: Experimental Mathematics
publication_status: published
publisher: Taylor & Francis
publist_id: '5070'
quality_controlled: '1'
scopus_import: 1
status: public
title: The monotone secant conjecture in the real Schubert calculus
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 24
year: '2015'
...
---
_id: '1579'
abstract:
- lang: eng
text: We show that the Galois group of any Schubert problem involving lines in projective
space contains the alternating group. This constitutes the largest family of enumerative
problems whose Galois groups have been largely determined. Using a criterion of
Vakil and a special position argument due to Schubert, our result follows from
a particular inequality among Kostka numbers of two-rowed tableaux. In most cases,
a combinatorial injection proves the inequality. For the remaining cases, we use
the Weyl integral formulas to obtain an integral formula for these Kostka numbers.
This rewrites the inequality as an integral, which we estimate to establish the
inequality.
acknowledgement: "This research was supported in part by NSF grant DMS-915211 and
the Institut Mittag-Leffler.\r\n"
article_processing_charge: No
author:
- first_name: Christopher
full_name: Brooks, Christopher
last_name: Brooks
- first_name: Abraham
full_name: Martin Del Campo Sanchez, Abraham
id: 4CF47F6A-F248-11E8-B48F-1D18A9856A87
last_name: Martin Del Campo Sanchez
- first_name: Frank
full_name: Sottile, Frank
last_name: Sottile
citation:
ama: Brooks C, Martin del Campo Sanchez A, Sottile F. Galois groups of Schubert
problems of lines are at least alternating. Transactions of the American Mathematical
Society. 2015;367(6):4183-4206. doi:10.1090/S0002-9947-2014-06192-8
apa: Brooks, C., Martin del Campo Sanchez, A., & Sottile, F. (2015). Galois
groups of Schubert problems of lines are at least alternating. Transactions
of the American Mathematical Society. American Mathematical Society. https://doi.org/10.1090/S0002-9947-2014-06192-8
chicago: Brooks, Christopher, Abraham Martin del Campo Sanchez, and Frank Sottile.
“Galois Groups of Schubert Problems of Lines Are at Least Alternating.” Transactions
of the American Mathematical Society. American Mathematical Society, 2015.
https://doi.org/10.1090/S0002-9947-2014-06192-8.
ieee: C. Brooks, A. Martin del Campo Sanchez, and F. Sottile, “Galois groups of
Schubert problems of lines are at least alternating,” Transactions of the American
Mathematical Society, vol. 367, no. 6. American Mathematical Society, pp.
4183–4206, 2015.
ista: Brooks C, Martin del Campo Sanchez A, Sottile F. 2015. Galois groups of Schubert
problems of lines are at least alternating. Transactions of the American Mathematical
Society. 367(6), 4183–4206.
mla: Brooks, Christopher, et al. “Galois Groups of Schubert Problems of Lines Are
at Least Alternating.” Transactions of the American Mathematical Society,
vol. 367, no. 6, American Mathematical Society, 2015, pp. 4183–206, doi:10.1090/S0002-9947-2014-06192-8.
short: C. Brooks, A. Martin del Campo Sanchez, F. Sottile, Transactions of the American
Mathematical Society 367 (2015) 4183–4206.
date_created: 2018-12-11T11:52:50Z
date_published: 2015-06-01T00:00:00Z
date_updated: 2021-01-12T06:51:43Z
day: '01'
department:
- _id: CaUh
doi: 10.1090/S0002-9947-2014-06192-8
intvolume: ' 367'
issue: '6'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: http://arxiv.org/abs/1207.4280
month: '06'
oa: 1
oa_version: Preprint
page: 4183 - 4206
publication: Transactions of the American Mathematical Society
publication_status: published
publisher: American Mathematical Society
publist_id: '5592'
quality_controlled: '1'
scopus_import: 1
status: public
title: Galois groups of Schubert problems of lines are at least alternating
type: journal_article
user_id: 3E5EF7F0-F248-11E8-B48F-1D18A9856A87
volume: 367
year: '2015'
...
---
_id: '2178'
abstract:
- lang: eng
text: We consider the three-state toric homogeneous Markov chain model (THMC) without
loops and initial parameters. At time T, the size of the design matrix is 6 ×
3 · 2T-1 and the convex hull of its columns is the model polytope. We study the
behavior of this polytope for T ≥ 3 and we show that it is defined by 24 facets
for all T ≥ 5. Moreover, we give a complete description of these facets. From
this, we deduce that the toric ideal associated with the design matrix is generated
by binomials of degree at most 6. Our proof is based on a result due to Sturmfels,
who gave a bound on the degree of the generators of a toric ideal, provided the
normality of the corresponding toric variety. In our setting, we established the
normality of the toric variety associated to the THMC model by studying the geometric
properties of the model polytope.
acknowledgement: Research of Martín del Campo supported in part by NSF Grant DMS-915211.
author:
- first_name: David
full_name: Haws, David
last_name: Haws
- first_name: Abraham
full_name: Martin Del Campo Sanchez, Abraham
id: 4CF47F6A-F248-11E8-B48F-1D18A9856A87
last_name: Martin Del Campo Sanchez
- first_name: Akimichi
full_name: Takemura, Akimichi
last_name: Takemura
- first_name: Ruriko
full_name: Yoshida, Ruriko
last_name: Yoshida
citation:
ama: Haws D, Martin del Campo Sanchez A, Takemura A, Yoshida R. Markov degree of
the three-state toric homogeneous Markov chain model. Beitrage zur Algebra
und Geometrie. 2014;55(1):161-188. doi:10.1007/s13366-013-0178-y
apa: Haws, D., Martin del Campo Sanchez, A., Takemura, A., & Yoshida, R. (2014).
Markov degree of the three-state toric homogeneous Markov chain model. Beitrage
Zur Algebra Und Geometrie. Springer. https://doi.org/10.1007/s13366-013-0178-y
chicago: Haws, David, Abraham Martin del Campo Sanchez, Akimichi Takemura, and Ruriko
Yoshida. “Markov Degree of the Three-State Toric Homogeneous Markov Chain Model.”
Beitrage Zur Algebra Und Geometrie. Springer, 2014. https://doi.org/10.1007/s13366-013-0178-y.
ieee: D. Haws, A. Martin del Campo Sanchez, A. Takemura, and R. Yoshida, “Markov
degree of the three-state toric homogeneous Markov chain model,” Beitrage zur
Algebra und Geometrie, vol. 55, no. 1. Springer, pp. 161–188, 2014.
ista: Haws D, Martin del Campo Sanchez A, Takemura A, Yoshida R. 2014. Markov degree
of the three-state toric homogeneous Markov chain model. Beitrage zur Algebra
und Geometrie. 55(1), 161–188.
mla: Haws, David, et al. “Markov Degree of the Three-State Toric Homogeneous Markov
Chain Model.” Beitrage Zur Algebra Und Geometrie, vol. 55, no. 1, Springer,
2014, pp. 161–88, doi:10.1007/s13366-013-0178-y.
short: D. Haws, A. Martin del Campo Sanchez, A. Takemura, R. Yoshida, Beitrage Zur
Algebra Und Geometrie 55 (2014) 161–188.
date_created: 2018-12-11T11:56:10Z
date_published: 2014-03-01T00:00:00Z
date_updated: 2021-01-12T06:55:48Z
day: '01'
department:
- _id: CaUh
doi: 10.1007/s13366-013-0178-y
intvolume: ' 55'
issue: '1'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: http://arxiv.org/abs/1204.3070
month: '03'
oa: 1
oa_version: Submitted Version
page: 161 - 188
publication: Beitrage zur Algebra und Geometrie
publication_status: published
publisher: Springer
publist_id: '4804'
quality_controlled: '1'
scopus_import: 1
status: public
title: Markov degree of the three-state toric homogeneous Markov chain model
type: journal_article
user_id: 4435EBFC-F248-11E8-B48F-1D18A9856A87
volume: 55
year: '2014'
...
---
_id: '5920'
abstract:
- lang: eng
text: We study chains of lattice ideals that are invariant under a symmetric group
action. In our setting, the ambient rings for these ideals are polynomial rings
which are increasing in (Krull) dimension. Thus, these chains will fail to stabilize
in the traditional commutative algebra sense. However, we prove a theorem which
says that “up to the action of the group”, these chains locally stabilize. We
also give an algorithm, which we have implemented in software, for explicitly
constructing these stabilization generators for a family of Laurent toric ideals
involved in applications to algebraic statistics. We close with several open problems
and conjectures arising from our theoretical and computational investigations.
article_processing_charge: No
article_type: original
author:
- first_name: Christopher J.
full_name: Hillar, Christopher J.
last_name: Hillar
- first_name: Abraham
full_name: Martin del Campo Sanchez, Abraham
id: 4CF47F6A-F248-11E8-B48F-1D18A9856A87
last_name: Martin del Campo Sanchez
citation:
ama: Hillar CJ, Martin del Campo Sanchez A. Finiteness theorems and algorithms for
permutation invariant chains of Laurent lattice ideals. Journal of Symbolic
Computation. 2013;50:314-334. doi:10.1016/j.jsc.2012.06.006
apa: Hillar, C. J., & Martin del Campo Sanchez, A. (2013). Finiteness theorems
and algorithms for permutation invariant chains of Laurent lattice ideals. Journal
of Symbolic Computation. Elsevier. https://doi.org/10.1016/j.jsc.2012.06.006
chicago: Hillar, Christopher J., and Abraham Martin del Campo Sanchez. “Finiteness
Theorems and Algorithms for Permutation Invariant Chains of Laurent Lattice Ideals.”
Journal of Symbolic Computation. Elsevier, 2013. https://doi.org/10.1016/j.jsc.2012.06.006.
ieee: C. J. Hillar and A. Martin del Campo Sanchez, “Finiteness theorems and algorithms
for permutation invariant chains of Laurent lattice ideals,” Journal of Symbolic
Computation, vol. 50. Elsevier, pp. 314–334, 2013.
ista: Hillar CJ, Martin del Campo Sanchez A. 2013. Finiteness theorems and algorithms
for permutation invariant chains of Laurent lattice ideals. Journal of Symbolic
Computation. 50, 314–334.
mla: Hillar, Christopher J., and Abraham Martin del Campo Sanchez. “Finiteness Theorems
and Algorithms for Permutation Invariant Chains of Laurent Lattice Ideals.” Journal
of Symbolic Computation, vol. 50, Elsevier, 2013, pp. 314–34, doi:10.1016/j.jsc.2012.06.006.
short: C.J. Hillar, A. Martin del Campo Sanchez, Journal of Symbolic Computation
50 (2013) 314–334.
date_created: 2019-02-05T08:48:24Z
date_published: 2013-03-01T00:00:00Z
date_updated: 2021-01-12T08:05:15Z
day: '01'
doi: 10.1016/j.jsc.2012.06.006
extern: '1'
intvolume: ' 50'
language:
- iso: eng
month: '03'
oa_version: None
page: 314-334
publication: Journal of Symbolic Computation
publication_identifier:
issn:
- 0747-7171
publication_status: published
publisher: Elsevier
quality_controlled: '1'
related_material:
link:
- relation: erratum
url: https://doi.org/10.1016/j.jsc.2015.09.002
status: public
title: Finiteness theorems and algorithms for permutation invariant chains of Laurent
lattice ideals
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 50
year: '2013'
...