Rojas, J. Maurice (Joseph Maurice)
Overview
Works:  9 works in 22 publications in 1 language and 902 library holdings 

Genres:  Conference papers and proceedings 
Roles:  Editor, Author, Thesis advisor 
Classifications:  QA297, 004.0151 
Publication Timeline
.
Most widely held works by
J. Maurice Rojas
Foundations of computational mathematics : proceedings of the Smalefest 2000, Hong Kong, 1317, 2000 by
Felipe Cucker(
Book
)
12 editions published in 2002 in English and Undetermined and held by 72 WorldCat member libraries worldwide
This invaluable book contains 19 papers selected from those submitted to a conference held in Hong Kong in July 2000 to celebrate the 70th birthday of Professor Steve Smale. It may be regarded as a continuation of the proceedings of SMALEFEST 1990 ("From Topology to Computation") held in Berkeley, USA, 10 years before, but with the focus on the area in which Smale worked more intensively during the '90's, namely the foundations of computational mathematics
12 editions published in 2002 in English and Undetermined and held by 72 WorldCat member libraries worldwide
This invaluable book contains 19 papers selected from those submitted to a conference held in Hong Kong in July 2000 to celebrate the 70th birthday of Professor Steve Smale. It may be regarded as a continuation of the proceedings of SMALEFEST 1990 ("From Topology to Computation") held in Berkeley, USA, 10 years before, but with the focus on the area in which Smale worked more intensively during the '90's, namely the foundations of computational mathematics
Cohomology, combinatorics, and complexity arising from solving polynomial systems by
J. Maurice Rojas(
)
2 editions published in 1995 in English and held by 5 WorldCat member libraries worldwide
2 editions published in 1995 in English and held by 5 WorldCat member libraries worldwide
Optimal condition for determining the exact number of roots of a polynomial system by
J. Maurice Rojas(
)
2 editions published in 1991 in English and held by 5 WorldCat member libraries worldwide
2 editions published in 1991 in English and held by 5 WorldCat member libraries worldwide
ADiscriminant Varieties and Amoebae by Korben Allen Rusek(
)
1 edition published in 2013 in English and held by 1 WorldCat member library worldwide
The motivating question behind this body of research is Smale's 17th problem: Can a zero of n complex polynomial equations in n unknowns be found approximately, on the average, in polynomial time with a uniform algorithm? While certain aspects and viewpoints of this problem have been solved, the analog over the real numbers largely remains open. This is an important question with applications in celestial mechanics, kinematics, polynomial optimization, and many others. Let A = {[alpha]₁, . . . , [alpha]_n+k } [subset] Z^n. The Adiscriminant variety is, among other things, a tool that can be used to categorize polynomials based on the topology of their real solution set. This fact has made it useful in solving aspects and special cases of Smale's 17th problem. In this thesis, we take a closer look at the structure of the Adiscriminant with an eye toward furthering progress on analogs of Smale's 17th problem. We examine a mostly ignored form called the Horn uniformization. This represents the discriminant in a compact form. We study properties of the Horn uniformization to find structural properties that can be used to better understand the Adiscriminant variety. In particular, we use a little known theorem of Kapranov limiting the normals of the Adiscriminant amoeba. We give new O(n²) bounds on the number of components in the complement of the real Adiscriminant when k = 3, where previous bounds had been O(n⁶) or even exponential before that. We introduce new tools that can be used in discovering various types of extremal Adiscriminants as well as examples found with these tools: a family of Adiscriminant varieties with the maximal number of cusps and a family that appears to asymptotically admit the maximal number of chambers. Finally we give sage code that efficiently plots the Adiscriminant amoeba for k = 3. Then we switch to a nonArchimedean point of view. Here we also give O(n²) bounds for the number of connected components in the complement of the non Archimedean Adiscriminant amoeba when k = 3, but we also get a bound of O(n²^(k1)(k2)) )when k >̲ 3. As in the real case, we also give a family exhibiting O(n²) connected components asymptotically. Finally we give code that efficiently plots the padic Adiscriminant amoeba for all k >̲ 3. These results help us understand the structure of the Adiscriminant to a degree, as yet, unknown. This can ultimately help in solving Smale's 17th problem as it gives a better understanding of how complicated the solution set can be. The electronic version of this dissertation is accessible from http://hdl.handle.net/1969.1/150998
1 edition published in 2013 in English and held by 1 WorldCat member library worldwide
The motivating question behind this body of research is Smale's 17th problem: Can a zero of n complex polynomial equations in n unknowns be found approximately, on the average, in polynomial time with a uniform algorithm? While certain aspects and viewpoints of this problem have been solved, the analog over the real numbers largely remains open. This is an important question with applications in celestial mechanics, kinematics, polynomial optimization, and many others. Let A = {[alpha]₁, . . . , [alpha]_n+k } [subset] Z^n. The Adiscriminant variety is, among other things, a tool that can be used to categorize polynomials based on the topology of their real solution set. This fact has made it useful in solving aspects and special cases of Smale's 17th problem. In this thesis, we take a closer look at the structure of the Adiscriminant with an eye toward furthering progress on analogs of Smale's 17th problem. We examine a mostly ignored form called the Horn uniformization. This represents the discriminant in a compact form. We study properties of the Horn uniformization to find structural properties that can be used to better understand the Adiscriminant variety. In particular, we use a little known theorem of Kapranov limiting the normals of the Adiscriminant amoeba. We give new O(n²) bounds on the number of components in the complement of the real Adiscriminant when k = 3, where previous bounds had been O(n⁶) or even exponential before that. We introduce new tools that can be used in discovering various types of extremal Adiscriminants as well as examples found with these tools: a family of Adiscriminant varieties with the maximal number of cusps and a family that appears to asymptotically admit the maximal number of chambers. Finally we give sage code that efficiently plots the Adiscriminant amoeba for k = 3. Then we switch to a nonArchimedean point of view. Here we also give O(n²) bounds for the number of connected components in the complement of the non Archimedean Adiscriminant amoeba when k = 3, but we also get a bound of O(n²^(k1)(k2)) )when k >̲ 3. As in the real case, we also give a family exhibiting O(n²) connected components asymptotically. Finally we give code that efficiently plots the padic Adiscriminant amoeba for all k >̲ 3. These results help us understand the structure of the Adiscriminant to a degree, as yet, unknown. This can ultimately help in solving Smale's 17th problem as it gives a better understanding of how complicated the solution set can be. The electronic version of this dissertation is accessible from http://hdl.handle.net/1969.1/150998
Some new applications of toric geometry by
J. Maurice Rojas(
Book
)
1 edition published in 1997 in English and held by 0 WorldCat member libraries worldwide
1 edition published in 1997 in English and held by 0 WorldCat member libraries worldwide
Toric generalized characteristic polynomials by
J. Maurice Rojas(
Book
)
1 edition published in 1997 in English and held by 0 WorldCat member libraries worldwide
1 edition published in 1997 in English and held by 0 WorldCat member libraries worldwide
Randomization, Relaxation, and Complexity in Polynomial Equation Solving by
Leonid Gurvits(
)
1 edition published in 2011 in English and held by 0 WorldCat member libraries worldwide
1 edition published in 2011 in English and held by 0 WorldCat member libraries worldwide
Extensions and corrections for "A convex geometric approach to counting the roots of a polynomial system" by
J. Maurice Rojas(
Book
)
1 edition published in 1996 in English and held by 0 WorldCat member libraries worldwide
1 edition published in 1996 in English and held by 0 WorldCat member libraries worldwide
Toric intersection theory for affine root counting by
J. Maurice Rojas(
Book
)
1 edition published in 1996 in English and held by 0 WorldCat member libraries worldwide
1 edition published in 1996 in English and held by 0 WorldCat member libraries worldwide
Audience Level
0 

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Related Identities
 Cucker, Felipe 1958 Author Editor
 Smale, Stephen 1930
 Pébay, Philippe
 Gurvits, Leonid Author
 Editors Editor
 Smale, Steve
 Texas A & M University
 Rusek, Korben Allen Author
 World Scientific (Singapur) Publisher
Associated Subjects