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BEGIN:VEVENT
SUMMARY:Petter Braenden (KTH Royal Institute of Technology)
DTSTART:20211018T132000Z
DTEND:20211018T141000Z
DTSTAMP:20260422T185239Z
UID:CMO-21w5117/1
DESCRIPTION:Title: Stable polynomials and related families of polynomials\nby Petter
Braenden (KTH Royal Institute of Technology) as part of CMO- Real Polynomi
als: Counting and Stability\n\n\nAbstract\nI will give a panoramic talk on
stable polynomials and related families of polynomials\, such as hyperbol
ic and Lorentzian polynomials. Over the past two decades stable polynomial
s and their relatives have been applied in different areas such as optimiz
ation\, real algebraic geometry\, combinatorics\, statistical mechanics\,
quantum mechanics and computer science. I will review some remarkable prop
erties of this class of polynomials as well as point to applications. I wi
ll also talk about a recent generalization called Lorentzian polynomials.\
n
LOCATION:https://researchseminars.org/talk/CMO-21w5117/1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Frédéric Bihan (Universite Savoie Mont Blanc)
DTSTART:20211018T143000Z
DTEND:20211018T152000Z
DTSTAMP:20260422T185239Z
UID:CMO-21w5117/2
DESCRIPTION:Title: Fewnomial bounds and multivariate generalisations of Descartes’ rule
of signs\nby Frédéric Bihan (Universite Savoie Mont Blanc) as part
of CMO- Real Polynomials: Counting and Stability\n\n\nAbstract\nIn 1980\,
A. Khovanskii gave a bound for the number of non-degenerate positive solut
ions of any square sparse polynomial system. His bound depends only on the
number of monomials of the system and is smaller than all classical bound
s (Bézout or mixed volume bounds) when the number of monomials is small.
Such bounds are called fewnomial bounds. In the univariate case\, the clas
sical Descartes’ rule of signs\, going back from 1637\, produces a bound
for the number of positive roots which takes care of the signs of the coe
fficients\, which is sharp and which implies a sharp fewnomial bound. In t
his talk\, I will describe several improvements of Khovanskii bound\, whic
h in some cases provide sharp fewnomial bounds. I will also describe recen
t multivariate generalisations of Descartes’ rule of signs. This talk is
mainly based on joint works with several collaborators including A. Dicke
nstein\, B. El Hilany\, J. Forsgard\, M. Rojas and F. Sottile.\n
LOCATION:https://researchseminars.org/talk/CMO-21w5117/2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Thorsten Theobald (Goethe-Universität Frankfurt/Main)
DTSTART:20211018T154000Z
DTEND:20211018T163000Z
DTSTAMP:20260422T185239Z
UID:CMO-21w5117/3
DESCRIPTION:Title: Conic stability of polynomials\, imaginary projections and spectrahedr
a\nby Thorsten Theobald (Goethe-Universität Frankfurt/Main) as part o
f CMO- Real Polynomials: Counting and Stability\n\n\nAbstract\nA multivari
ate polynomial $p$ in ${\\mathbb C}[z_1\, \\ldots\, z_n]$\nis called stabl
e if every root $z$ has at least one\ncomponent $z_j$ with imaginary part
$\\le 0$. In this\nexpository talk\, we discuss the naturally generalized\
nviewpoint of conic stability. Its origin is in the\nstudy of imaginary pr
ojections\, and the usual stability\nrefers to the specific polyhedral con
e ${\\mathbb R}_+^n$.\n\nAs a prominent case\, we consider conic stability
with\nrespect to the positive semidefinite cone ("psd stability").\nCrite
ria for psd stability are tightly linked to the\ncontainment problem for s
pectrahedra\, to positive maps\nand to determinantal representations.\n\nT
he own results in this talk are based on various joint\nworks with Giulia
Codenotti\, Papri Dey\, Stephan Gardoll\,\nThorsten Jörgens\, Mahsa Sayya
ry and Timo de Wolff.\n
LOCATION:https://researchseminars.org/talk/CMO-21w5117/3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Boris Shapiro (University of Stockholm)
DTSTART:20211019T130000Z
DTEND:20211019T134000Z
DTSTAMP:20260422T185239Z
UID:CMO-21w5117/4
DESCRIPTION:Title: Return of the plane evolute\nby Boris Shapiro (University of Stock
holm) as part of CMO- Real Polynomials: Counting and Stability\n\n\nAbstra
ct\nWe consider the evolutes of plane real-algebraic curves and discuss so
me of their complex and real-algebraic properties. In particular\, for a g
iven degree d ≥ 2\, we provide lower bounds for the following four numer
ical invariants: 1) the maximal number of times a real line can intersect
the evolute of a real-algebraic curve of degree d\; 2) the maximal number
of real cusps which can occur on the evolute of a real-algebraic curve of
degree d\; 3) the maximal number of (cru)nodes which can occur on the dual
curve to the evolute of a real-algebraic curve of degree d\; 4) the maxim
al number of (cru)nodes which can occur on the evolute of a real-algebraic
curve of degree d (joint with R.Piene and C.Riener).\n
LOCATION:https://researchseminars.org/talk/CMO-21w5117/4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Cristhian Garay López (CIMAT)
DTSTART:20211019T140000Z
DTEND:20211019T144000Z
DTSTAMP:20260422T185239Z
UID:CMO-21w5117/5
DESCRIPTION:Title: Inflection polynomials of linear series on superelliptic curves\nb
y Cristhian Garay López (CIMAT) as part of CMO- Real Polynomials: Countin
g and Stability\n\n\nAbstract\nWe explore the inflectionary behavior of li
near series on families of marked superelliptic curves (i.e.\, cyclic cove
rs of $\\mathbb{P}^1$). The inflection of these linear series supported a
way from the superelliptic ramification locus is parameterized by the infl
ection polynomials\, a certain family of polynomials generalizing the div
ision polynomials (which are used to compute the torsion points of ellipti
c curves). These polynomials are remarkable since their properties reflect
aspects of the underlying family of superelliptic curves. We also obtain
inflectionary varieties\, which describe the global behaviour of the infle
ction points on the family.\n\nIn this talk we will introduce these inflec
tion polynomials and some of their properties. Although this story is vali
d over fields of arbitrary characteristic\, we will focus on the real case
. \nWe report on joint work with Ethan Cotterill\, Ignacio Darago\, Changh
o Han\, and\nTony Shaska.\n
LOCATION:https://researchseminars.org/talk/CMO-21w5117/5/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mareike Dressler (UC San Diego)
DTSTART:20211019T150000Z
DTEND:20211019T154000Z
DTSTAMP:20260422T185239Z
UID:CMO-21w5117/6
DESCRIPTION:Title: Real zeros of sums of nonnegative circuit polynomials\nby Mareike
Dressler (UC San Diego) as part of CMO- Real Polynomials: Counting and Sta
bility\n\n\nAbstract\nUnderstanding the real zeros of polynomials is a res
earch subject of intrinsic interest with a long and rich history and is es
pecially useful for polynomials with specific properties like nonnegativit
y. In this talk\, I provide a complete and explicit characterization of th
e real zeros of both homogeneous and inhomogeneous sums of nonnegative cir
cuit (SONC) polynomials\, a recent certificate for nonnegative polynomials
independent of sums of squares. As an interesting consequence\, I show th
at the supremum of the number of zeros of all homogeneous n-variate polyn
omials of degree 2d in the SONC cone can be determined exactly. Note that
in strong contrast\, the determination of this number for both the nonneg
ativity cone and the cone of sums of squares for general n and d is st
ill an open question.\n
LOCATION:https://researchseminars.org/talk/CMO-21w5117/6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Cynthia Vinzant (University of Washington)
DTSTART:20211019T160000Z
DTEND:20211019T164000Z
DTSTAMP:20260422T185239Z
UID:CMO-21w5117/7
DESCRIPTION:Title: Log-concavity and applications to approximate counting and sampling in
matroids\nby Cynthia Vinzant (University of Washington) as part of CM
O- Real Polynomials: Counting and Stability\n\n\nAbstract\nMatroids are co
mbinatorial objects designed to capture independence relations on collecti
ons of objects\, such as linear independence of vectors in a vector space
or cyclic independence of edges in a graph. Recent work by several indepen
dent authors shows that the multivariate basis-generating polynomial of a
matroid is log-concave as a function on the positive orthant. In this talk
\, I will describe some of the underlying combinatorial and geometric stru
cture of such log-concave polynomials and applications to the problems of
approximately counting and approximately sampling the bases of a matroid.
This is based on joint work with Nima Anari\, Kuikui Liu\, and Shayan Ovei
s Gharan.\n
LOCATION:https://researchseminars.org/talk/CMO-21w5117/7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Claus Scheiderer (Univ-Konstanz Germany)
DTSTART:20211020T130000Z
DTEND:20211020T134000Z
DTSTAMP:20260422T185239Z
UID:CMO-21w5117/8
DESCRIPTION:Title: Low-complexity semidefinite representation of convex hulls of curves\nby Claus Scheiderer (Univ-Konstanz Germany) as part of CMO- Real Polyn
omials: Counting and Stability\n\n\nAbstract\nMatroids are combinatorial o
bjects designed to capture independence relations on collections of object
s\, such as linear independence of vectors in a vector space or cyclic ind
ependence of edges in a graph. Recent work by several independent authors
shows that the multivariate basis-generating polynomial of a matroid is lo
g-concave as a function on the positive orthant. In this talk\, I will des
cribe some of the underlying combinatorial and geometric structure of such
log-concave polynomials and applications to the problems of approximately
counting and approximately sampling the bases of a matroid. This is based
on joint work with Nima Anari\, Kuikui Liu\, and Shayan Oveis Gharan.\n
LOCATION:https://researchseminars.org/talk/CMO-21w5117/8/
END:VEVENT
BEGIN:VEVENT
SUMMARY:El Hilany Boulos (TU Braunschweig)
DTSTART:20211020T140000Z
DTEND:20211020T144000Z
DTSTAMP:20260422T185239Z
UID:CMO-21w5117/9
DESCRIPTION:Title: A polyhedral description for the non-properness set of a polynomial ma
p\nby El Hilany Boulos (TU Braunschweig) as part of CMO- Real Polynomi
als: Counting and Stability\n\n\nAbstract\nLet $K$ be the field of either
real or complex numbers\, and let $S_f$ denote the set of points in $K^n$
at which a polynomial map $f: K^n\\to K^n$ is not proper.\nJelonek proved
that $S_f$ is an algebraic hypersurface in the complex case and semi-algeb
raic in the real case. He furthermore showed that $S_f$ is ruled by polyno
mial\ncurves\, and provided a method for computing its equations for compl
ex maps. However\, such methods do not extend to real polynomial maps.\n\n
In this talk\, I will establish a description of $S_f$ for\na large family
of non-proper polynomial maps f using their Newton polytopes. I will furt
hermore highlight the interplay between the combinatorics of the polytopes
and the topology of $S_f$. The resulting method computes $S_f$ for comple
x polynomial maps as well as the real ones. As an application\, some of\nJ
elonek's results are recovered. \n\nI will furthermore report on a joint
work with Elias Tsigaridas in which we provided a more elaborate method fo
r computing the non-properness set for degenerate real polynomial maps on
the plane.\n
LOCATION:https://researchseminars.org/talk/CMO-21w5117/9/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mario Kummer (TU Berlin Germany)
DTSTART:20211020T150000Z
DTEND:20211020T154000Z
DTSTAMP:20260422T185239Z
UID:CMO-21w5117/10
DESCRIPTION:Title: Matroids with the half-plane property and related concepts\nby Ma
rio Kummer (TU Berlin Germany) as part of CMO- Real Polynomials: Counting
and Stability\n\n\nAbstract\nWe will study several properties of bases gen
erating polynomials of matroids that are related to stability. This includ
es the half-plane property or determinantal representability. We will furt
her present a classification of all matroids on up to eight elements whose
bases generating polynomial is stable. This is joint work with Büşra Se
rt.\n
LOCATION:https://researchseminars.org/talk/CMO-21w5117/10/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Papri Day (University of Missouri)
DTSTART:20211020T160000Z
DTEND:20211020T164000Z
DTSTAMP:20260422T185239Z
UID:CMO-21w5117/11
DESCRIPTION:Title: Real Degeneracy Loci of Matrices\, and Hyperbolicity cones of Real Po
lynomials\nby Papri Day (University of Missouri) as part of CMO- Real
Polynomials: Counting and Stability\n\n\nAbstract\nThis talk has two parts
. In the first part\, I shall talk about real degeneracy loci of matrices
and its correspondence with symmetroids. Let $\\mathcal{A}:=\\{A_1 \\dots\
,A_{m+1}\\}$ be a collection of linear operators on ${\\mathbb R}^{m}$. Th
e degeneracy locus (DL) of $\\mathcal{A}$ is defined as the set of the poi
nts $x$ for which rank$([A_1x\\dots A_{m+1}x])\\leq m-1$. We show that the
DL is an $m-3$ dimensional sub-scheme of degree ${m+1 \\choose 2}$ in ${\
\mathbb P}^{m-1}({\\mathbb C})$. In particular\, when $m=3$\, the DL consi
sts of six rational points in ${\\mathbb P}^{2}({\\mathbb R})$ with quadri
lateral configuration if and only if $A_{i}\,i=1\\dots\,4$ are in the line
ar span of four fixed rank-one operators. Moreover\, we show that if $A_{i
}\,i=1\\dots\,m+1$ are in the linear span of $m+1$ fixed rank-one matrices
\, the DL of $m+1$ matrices satisfies generalized Desargues configuration\
, and it has correspondence with a Special type of symmetroid\, call it Sy
lvester symmetroid. This part is based on joint work with Dan Edidin.\n\nI
n the second part\, I shall focus on the hyperbolicity cones of the elemen
tary symmetric polynomials and real polynomials which define symmetroids (
work in progress).\n
LOCATION:https://researchseminars.org/talk/CMO-21w5117/11/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mahsa Sayyary Namin (Goethe University Frankfurt)
DTSTART:20211021T130000Z
DTEND:20211021T134000Z
DTSTAMP:20260422T185239Z
UID:CMO-21w5117/12
DESCRIPTION:Title: Imaginary Projections: Complex Versus Real Coefficients\nby Mahsa
Sayyary Namin (Goethe University Frankfurt) as part of CMO- Real Polynomi
als: Counting and Stability\n\n\nAbstract\nGiven a complex multivariate po
lynomial \n${p\\in\\mathbb{C}[z_1\,\\ldots\,z_n]}$\, the imaginary project
ion \n$\\mathcal{I}(p)$ of $p$ is defined as the projection of the variety
\n$\\mathcal{V}(p)$ onto its imaginary part. We give a full \ncharacteriz
ation of the imaginary projections of conic sections with \ncomplex coeffi
cients\, which generalizes a classification for the case of \nreal conics.
More precisely\, given a bivariate complex polynomial \n$p\\in\\mathbb{C}
[z_1\,z_2]$ of total degree two\, we describe the number \nand the bounded
ness of the components in the complement of \n$\\mathcal{I}(p)$ as well as
their boundary curves and the spectrahedral \nstructure of the components
. We further study the imaginary projections \nof some families of higher
degree complex polynomials.\n
LOCATION:https://researchseminars.org/talk/CMO-21w5117/12/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mauricio Velasco (Universidad de los Andes)
DTSTART:20211021T140000Z
DTEND:20211021T144000Z
DTSTAMP:20260422T185239Z
UID:CMO-21w5117/13
DESCRIPTION:Title: Harmonic hierarchies for polynomial optimization\nby Mauricio Vel
asco (Universidad de los Andes) as part of CMO- Real Polynomials: Counting
and Stability\n\n\nAbstract\nThe cone of nonnegative forms of a given deg
ree is a convex set of remarkable beauty and usefulness.\nIn this talk we
will discuss some recent ideas for approximating this set through polyhedr
a and spectrahedra. We call the resulting approximations harmonic hierarch
ies since they arise naturally from harmonic analysis on spheres (or equiv
alently from the representation theory of $SO(n)$). We will describe theor
etical results leading to sharp estimates for the quality of these approxi
mations and will also show a brief demo of our Julia implementation of har
monic hierarchies. These results are joint work with Sergio Cristancho (Un
iAndes).\n
LOCATION:https://researchseminars.org/talk/CMO-21w5117/13/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Josephine Yu (Georgia Institute of Technology)
DTSTART:20211021T150000Z
DTEND:20211021T154000Z
DTSTAMP:20260422T185239Z
UID:CMO-21w5117/14
DESCRIPTION:Title: Positively Hyperbolic Varieties\, Tropicalization\, and Positroids\nby Josephine Yu (Georgia Institute of Technology) as part of CMO- Real
Polynomials: Counting and Stability\n\n\nAbstract\nWe will discuss a gener
alization of stable polynomials to complex algebraic varieties of codimens
ion larger than one and study their combinatorial structure using tropical
geometry. We show that their tropicalization are closely related to type-
A braid arrangements and positroids (matroid arising from the nonnegative
part of the Grassmannian) and that their Chow polytopes are generalized pe
rmutohedra. This is based on joint work with Felipe Rincón and Cynthia Vi
nzant.\n
LOCATION:https://researchseminars.org/talk/CMO-21w5117/14/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Máté László Telek (University of Copenhagen)
DTSTART:20211021T155000Z
DTEND:20211021T160500Z
DTSTAMP:20260422T185239Z
UID:CMO-21w5117/15
DESCRIPTION:Title: On generalizing Descartes' rule of signs to hypersurfaces\nby Má
té László Telek (University of Copenhagen) as part of CMO- Real Polynom
ials: Counting and Stability\n\n\nAbstract\nWe provide upper bounds on the
number of connected components of the complement of a hypersurface in the
positive orthant and phrase our results as partial generalizations of the
classical Descartes’ rule of signs to multivariate polynomials (with re
al exponents). In particular\, we give conditions based on the geometrical
configuration of the exponents and the sign of the coefficients that guar
antee that the number of connected components of the complement of the hyp
ersurface where the defining polynomial attains a negative value is at mos
t one or two. Furthermore\, we briefly present an application for chemical
reaction networks that motivated us to consider this problem.\n
LOCATION:https://researchseminars.org/talk/CMO-21w5117/15/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Abeer Al Ahmadieh (University of Washington)
DTSTART:20211021T160500Z
DTEND:20211021T162000Z
DTSTAMP:20260422T185239Z
UID:CMO-21w5117/16
DESCRIPTION:Title: Determinantal Representations and the Image of the Principal Minor Ma
p\nby Abeer Al Ahmadieh (University of Washington) as part of CMO- Rea
l Polynomials: Counting and Stability\n\n\nAbstract\nThe principal minor m
ap takes an $n$ by $n$ square matrix to the length-$2^n$ vector of its p
rincipal minors. A basic question is to give necessary and sufficient cond
itions that characterize the image of various spaces of matrices under thi
s map. In this talk I will describe the image of the space of complex matr
ices using a characterization of determinantal representations of multiaff
ine polynomials\, based on the factorization of their Rayleigh differences
. Using these techniques I will give equations and inequalities characteri
zing the images of the spaces of real and complex symmetric\, Hermitian\,
and general complex matrices. For complex symmetric matrices this recovers
a result of Oeding from $2011$. This is based on a joint work with Cynthi
a Vinzant.\n
LOCATION:https://researchseminars.org/talk/CMO-21w5117/16/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Cédric Le Texier (Oslo University)
DTSTART:20211021T162000Z
DTEND:20211021T163500Z
DTSTAMP:20260422T185239Z
UID:CMO-21w5117/17
DESCRIPTION:Title: Hyperbolic plane curves near the non-singular tropical limit\nby
Cédric Le Texier (Oslo University) as part of CMO- Real Polynomials: Coun
ting and Stability\n\n\nAbstract\nWe develop tools of real tropical inters
ection theory in order to determine necessary and sufficient conditions fo
r real algebraic curves near the non-singular tropical limit to be hyperbo
lic with respect to a point\, in terms of real phase structure and twisted
edges on a tropical curve\, generalising Speyer's classification of stabl
e curves near the tropical limit.\n
LOCATION:https://researchseminars.org/talk/CMO-21w5117/17/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Josué Tonelli-Cueto (Inria Paris & IMJ-PRG)
DTSTART:20211021T163500Z
DTEND:20211021T165000Z
DTSTAMP:20260422T185239Z
UID:CMO-21w5117/18
DESCRIPTION:Title: Metric restrictions on the number of real zeros\nby Josué Tonell
i-Cueto (Inria Paris & IMJ-PRG) as part of CMO- Real Polynomials: Counting
and Stability\n\n\nAbstract\nA well-known fact in real algebraic geometry
is that crossing the discriminant changes the number of real zeros. Howev
er\, can the size of a discriminant chamber influence the number of zeros
of the polynomial systems in it? In this talk\, we show some novel results
showing that this is the case. More concretely\, we show that we can boun
d the number of real zeros in terms of the logarithm of the inverse distan
ce to the discriminant—also known as the condition number—. We also de
monstrate that this bound has important consequences regarding random real
polynomial systems. This is joint work with Elias Tsigaridas.\n
LOCATION:https://researchseminars.org/talk/CMO-21w5117/18/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Khazhgali Kozhasov (Universität Osnabrück)
DTSTART:20211022T130000Z
DTEND:20211022T134000Z
DTSTAMP:20260422T185239Z
UID:CMO-21w5117/19
DESCRIPTION:Title: The many faces of polynomial capacity\nby Khazhgali Kozhasov (Uni
versität Osnabrück) as part of CMO- Real Polynomials: Counting and Stabi
lity\n\n\nAbstract\nThe capacity of a polynomial p with non-negative coeff
icients is a certain function on its support that interpolates between coe
fficients of p and its value at the vector (1\,...\,1). This concept has a
lot of remarkable applications including bounds on the mixed volume of co
nvex bodies and bounds on some combinatorial quantities like the number of
matchings in bipartite graphs. I will discuss relation of capacity to rel
ative entropy of measures as well as its appearances in the theory of non-
negative polynomials and in the theory of A-discriminants. The talk is bas
ed on a joint work in progress with Jonathan Leake and Timo de Wolff.\n
LOCATION:https://researchseminars.org/talk/CMO-21w5117/19/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Simone Naldi (Université de Limoges)
DTSTART:20211022T140000Z
DTEND:20211022T144000Z
DTSTAMP:20260422T185239Z
UID:CMO-21w5117/20
DESCRIPTION:Title: Spectrahedral representations of hyperbolic plane curves\nby Simo
ne Naldi (Université de Limoges) as part of CMO- Real Polynomials: Counti
ng and Stability\n\n\nAbstract\nA key question in the theory of hyperbolic
polynomials is how to test hyperbolicity. This is classically done by com
puting a symmetric determinantal representation of the given polynomial. I
n the case of curves this is always possible\, whereas in high dimension o
ne should look at such representations for multiples of the given polynomi
al (according to the Generalized Lax Conjecture). In the talk I will discu
ss a recent variant of the classical Dixon method\, for the computation of
spectrahedral representations of curves. The talk is based on a recent wo
rk joint with Mario Kummer and Daniel Plaumann.\n
LOCATION:https://researchseminars.org/talk/CMO-21w5117/20/
END:VEVENT
BEGIN:VEVENT
SUMMARY:J. Maurice Rojas (exas A & M University)
DTSTART:20211022T150000Z
DTEND:20211022T154000Z
DTSTAMP:20260422T185239Z
UID:CMO-21w5117/21
DESCRIPTION:Title: Counting Pieces of Real Near-Circuit Hypersurfaces Faster\nby J.
Maurice Rojas (exas A & M University) as part of CMO- Real Polynomials: Co
unting and Stability\n\n\nAbstract\nRandomization has proved instrumental
in efficiently solving\ngeometric problems where the best deterministic me
thods are impractical.\nAn important recent example is a recent singly exp
onential algorithm of\nBurgisser\, Cucker\, and Tonelli-Cueto for computin
g the homology of real\nalgebraic sets for ``most'' inputs. We approach an
analogous speed-up in a\ndifferent direction: Computing the isotopy type
of real zero sets\ndefined by certain n-variate sparse polynomials of degr
ee d with\ncoefficients of maximum bit-length h. We show how\,\nfor ``most
'' inputs\, we can compute the number of connected components\nof the posi
tive zero set in time $(h log d)^O(n)$\, whereas the fastest\nprevious alg
orithms had complexity $(hd)^{O(n)}$. A key tool is a new\nway to metrical
ly approximate certain A-discriminant varieties. We'll aslo\nsee how reduc
ing the dependence on the number of variables n is related\nto diophantine
approximation.\n\nParts of this work are joint with Frederic Bihan\, Jens
Forsgard\, Mounir\nNisse\, Kaitlyn Phillipson\, and Lisa Soule.\n
LOCATION:https://researchseminars.org/talk/CMO-21w5117/21/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Lucia Lopez de Medrano (Universidad Nacional Autonoma de Mexico)
DTSTART:20211022T160000Z
DTEND:20211022T164000Z
DTSTAMP:20260422T185239Z
UID:CMO-21w5117/22
DESCRIPTION:Title: On maximally inflected hyperbolic curves\nby Lucia Lopez de Medra
no (Universidad Nacional Autonoma de Mexico) as part of CMO- Real Polynomi
als: Counting and Stability\n\n\nAbstract\nIn this talk we will focus on t
he distribution of real inflection points among the ovals of a real non-si
ngular hyperbolic curve of even degree. Using Hilbert’s method we show t
hat for any integers $d$ and $r$ such that $4 ≤ r ≤ 2d^2 −2d$\, ther
e is a non-singular hyperbolic curve of degree $2d$ in $\\mathbb R^2$ with
exactly $r$ line segments in the boundary of its convex hull. We also giv
e a complete classification of possible distributions of inflection points
among the ovals of a maximally inflected non-singular hyperbolic curve of
degree 6. This is a joint work with Aubin Arroyo and Erwan Brugallé.\n
LOCATION:https://researchseminars.org/talk/CMO-21w5117/22/
END:VEVENT
END:VCALENDAR