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A characterization of the 0-basis homogeneous bounding degrees

2010
*
Journal of Symbolic Logic (JSL)
*

*The*result

*of*this paper shows that this is an exact

*characterization*

*of*

*the*

*0*-

*basis*

*homogeneous*

*bounding*Δ2

*0*

*degrees*. ... In previous work, we showed that

*the*non low2 Δ2

*0*

*degrees*are

*0*-

*basis*

*homogeneous*

*bounding*. ... We now show that

*the*

*0*-

*basis*

*homogeneous*

*bounding*

*degrees*exactly

*characterize*

*the*nonlow 2

*degrees*below

*0*. ...

##
###
The degree spectra of homogeneous models

2008
*
Journal of Symbolic Logic (JSL)
*

*A*

*degree*d is

*0*-

*homogeneous*

*bounding*if any automorphically nontrivial

*homogeneous*with

*a*

*0*-

*basis*has

*a*d-decidable copy. We show that

*the*nonlow2

*degrees*are

*0*-

*homogeneous*

*bounding*. ... In

*the*case where all types

*of*

*the*theory T are computable and is

*a*

*homogeneous*model with

*a*

*0*-

*basis*, we show has copies decidable in every nonzero

*degree*. ... They exactly

*characterized*

*the*

*homogeneous*

*bounding*

*degrees*as

*the*

*degrees*

*of*Peano arithmetic. ...

##
###
Computability of Homogeneous Models

2007
*
Notre Dame Journal of Formal Logic
*

*The*material in Lange [ta1] and [ta2] is part

*of*her Ph.D. dissertation for which Robert Soare and Denis Hirschfeldt are co-supervisers. ...

*The*first author presented some

*of*these and other results at an A.M.S. Special Session on October 23, 2004 at Northwestern University, and also at an A.M.S. meeting at Notre Dame, April 8-9, 2006. ...

*Degree*d is

*0*-

*bounding*if for any nontrivial

*homogeneous*model

*A*with

*a*

*0*-

*basis*, there exists

*a*d-decidable B ∼ =

*A*. ...

##
###
On the complexity of computing Gröbner bases for weighted homogeneous systems

2016
*
Journal of symbolic computation
*

Furthermore,

doi:10.1016/j.jsc.2015.12.001
fatcat:5xnhv5awu5dttgraorrc7dvo7m
*the*maximum*degree*reached in*a*run*of*Algorithm 5 is*bounded*by*the*weighted Macaulay*bound*∑ (d_i-w_i) + w_n, and this*bound*is sharp if we can order*the*weights so that w_n=1. ... Weighted*homogeneity*(or quasi-*homogeneity*) is one example*of*such*a*structure: given*a*system*of*weights W=(w_1,... ...*Degree**of*regularity*of**a*weighted*homogeneous*complete intersection*The**degree**of*regularity*of**a*zero-dimensional*homogeneous*regular system is*bounded*by Macaulay's*bound*and, in practice, that*bound*...##
###
Bases in spaces of homogeneous polynomials on Banach spaces

2007
*
Journal of Mathematical Analysis and Applications
*

In this work we present some conditions

doi:10.1016/j.jmaa.2006.09.058
fatcat:j53bp5byxjenvffhfgcrybcwju
*of*equivalence for*the*existence*of**a*monomial*basis*in spaces*of**homogeneous*polynomials on Banach spaces.*basis**of*c*0*is*a*polynomial-shrinking*basis*. ... If E = l 1 ,*the*canonical*basis*is not*a*polynomialshrinking*basis*, since it is not*a*shrinking*basis*. (b) Let E = d * (w, 1) be*the*predual*of*Lorentz's sequences space. ...*The*author wishes to acknowledge*the*financial support*of*FAPESP. ...##
###
On the Universal Gröbner bases of varieties of minimal degree

2008
*
Mathematical Research Letters
*

This leads to

doi:10.4310/mrl.2008.v15.n6.a11
fatcat:mgbe7y6yvrf33kcjhtiq2tobn4
*a*sharp*bound*on*the**degree**of*Graver*basis*elements, which is always attained by*a*circuit. ... Finally, for any variety obtained from*a*scroll by*a*sequence*of*projections to some*of**the*coordinate hyperplanes,*the**degree**of*any element in any reduced Gröbner*basis*is*bounded*by*the**degree**of**the*... Acknowledgments*The*author would like to thank Bernd Sturmfels for suggesting*a*generalization*of*primitive partition identities, and my advisor, Uwe Nagel, for his continuous support and guidance. ...##
###
Computing Minimal Generating Sets of Invariant Rings of Permutation Groups with SAGBI-Gröbner Basis

2001
*
Discrete Mathematics & Theoretical Computer Science
*

Our algorithm does not require

doi:10.46298/dmtcs.2285
fatcat:zrgiviiqyfclvpthedwsl5aame
*the*computation*of**a*Hironaka decomposition, nor even*the*computation*of**a*system*of*parameters, and could be parallelized. ... We circumvent*the*main weaknesses*of**the*usual approaches (using classical Gröbner*basis*inside*the*full polynomial ring, or pure linear algebra inside*the*invariant ring) by relying on*the*theory*of*SAGBI ... Acknowledgements We gratefully thank Lorenzo Robbiano and Jean-Charles Faugère for decisive ideas, as well as*the*Unité Mixte de Service Médicis, for providing*the*computational power. ...##
###
On the universal Gröbner bases of varieties of minimal degree
[article]

2007
*
arXiv
*
pre-print

Finally, for any variety obtained from

arXiv:0711.2714v2
fatcat:5m7owwmudjdfzcmefnvsbqms5u
*a*scroll by*a*sequence*of*projections to some*of**the*coordinate hyperplanes,*the**degree**of*any element in any reduced Gr\"obner*basis*is*bounded*by*the**degree**of*...*The*description*of**the*Graver bases is given in terms*of*colored partition identities. This leads to*a*sharp*bound*on*the**degree**of*Graver*basis*elements, which is always attained by*a*circuit. ...*The**degree**of*any binomial in*the*Graver*basis*(and*the*universal Gröbner*basis*)*of*any rational normal scroll is*bounded*above by*the**degree**of**the*scroll. ...##
###
Anisotropic Two-Microlocal Spaces and Regularity

2014
*
Journal of Function Spaces
*

We prove that these spaces allow

doi:10.1155/2014/505796
fatcat:s3wmp4agifge5ciaklqmnrio5y
*the**characterizing**of*pointwise anisotropic Hölder regularity. We also prove an anisotropic wavelet criterion for anisotropic uniform regularity. ... We finally prove that both this criterion and anisotropicD-u-two-microlocal spaces are independent*of**the*chosen anisotropicD-u-orthonormal wavelet*basis*. ... Acknowledgment*The*author would like to extend his sincere appreciation to*the*Deanship*of*Scientific Research at King Saud University for its funding*of*this research through*the*Research Group no. ...##
###
Homogenization and the Polynomial Calculus
[chapter]

2000
*
Lecture Notes in Computer Science
*

In standard implementations

doi:10.1007/3-540-45022-x_78
fatcat:bvzx63ozmfar3fjhd6ykbjrhwq
*of**the*Gröbner*basis*algorithm,*the*original polynomials are*homogenized*so that each term in*a*given polynomial has*the*same*degree*. ...*The*minimum PC refutation*degree**of**homogenized*formulas is equal to*the*Nullstellensatz refutation*degree**of**the*original formulas, whereas*the*size*of**the**homogenized*PC refutation is equal to*the*size ... Notice that*a*lower*bound*for zdeg(f ) is*a*lower*bound*for*the*power*of*Z which we can derive and hence is*a*lower*bound*on*the*HN-*degree*. ...##
###
Homogenization and the polynomial calculus

2002
*
Computational Complexity
*

In standard implementations

doi:10.1007/s00037-002-0171-6
fatcat:4wedhdl5obemtbrvwv2oosirzm
*of**the*Gröbner*basis*algorithm,*the*original polynomials are*homogenized*so that each term in*a*given polynomial has*the*same*degree*. ...*The*minimum PC refutation*degree**of**homogenized*formulas is equal to*the*Nullstellensatz refutation*degree**of**the*original formulas, whereas*the*size*of**the**homogenized*PC refutation is equal to*the*size ... Notice that*a*lower*bound*for zdeg(f ) is*a*lower*bound*for*the*power*of*Z which we can derive and hence is*a*lower*bound*on*the*HN-*degree*. ...##
###
Partition Theorems and Computability Theory

2005
*
Bulletin of Symbolic Logic
*

We denote

doi:10.2178/bsl/1122038995
fatcat:auevbp5frjgsplkxupnaqpbr6e
*the*set*of*natural numbers by ω and*the*set*of*finite sequences*of*natural numbers by ω<ω. We also identify each n ∈ ω with its set*of*predecessors, so n = {*0*, 1, 2, ..., n − 1}. ...*The*connections between mathematical logic and combinatorics have*a*rich history. ... Ramsey*Degrees*In Section 2, we saw that by shifting to an analysis*of**the*jumps*of**the**degrees**of**homogeneous*sets for computable f : [ω] 2 → 2, we were able to get tight*bounds*in*the*Turing*degrees*. ...##
###
The Degree of Regularity of HFE Systems
[chapter]

2010
*
Lecture Notes in Computer Science
*

Using this property with

doi:10.1007/978-3-642-17373-8_32
fatcat:a36u2tmygvfttiz2dykikxbk7q
*a*standard combinatorial calculation yields an arguably tight numerical*bound*on*the**degree**of*regularity*of*HFE systems for any parameters. ... More specifically, Faugère observed that*the*regular behaviour*of**the*Gröbner*basis*computation collapses at*a*much lower*degree*than expected for random systems, letting*the*computation finish much earlier ...*The*next paragraph is devoted to refining*the**characterization**of**the**degree**of*regularity*of*P*0*, . . . , P n−1 .*Characterizing**the**Degree**of*Regularity*of*Systems*of*R q n . ...##
###
On polynomial ideals, their complexity, and applications
[chapter]

1995
*
Lecture Notes in Computer Science
*

We discuss complexity results known for

doi:10.1007/3-540-60249-6_42
fatcat:soxs7hun4feqddpldpv2razz5y
*a*number*of*problems related to polynomial ideals, like*the*word problem for commutative semigroups,*a*quantitative version*of*Hilbert's Nullstellensatz, and*the*...*A*polynomial ideal membership problem is*a*(w+1)-tuple P = (f; g 1 ; g 2 ; : : : ; g w ) where f and*the*g i are multivariate polynomials over some ring, and*the*problem is to determine whether f is in ... m and*the*polynomials g j are*homogeneous*,*the*g j*of**degree*say 4 andm*0*?m*of**degree*roughly n. Now, M accepts m i m*0*? ...##
###
Some Complexity Results for Polynomial Ideals

1997
*
Journal of Complexity
*

In this paper, we survey some

doi:10.1006/jcom.1997.0447
fatcat:d3cz6pfw3za4disutcnitfvlsa
*of*our new results on*the*complexity*of**a*number*of*problems related to polynomial ideals. ... We discuss further complexity results for problems related to polynomial ideals, like*the*word and subword problems for commutative semigroups,*a*quantitative version*of*Hilbert's Nullstellensatz in*a*... . , x n ] be given, and let d be*a**bound*on*the*total*degree**of**the*g i . Then there is*a**basis*for*the*module*of*syzygies whose polynomials have*a*total*degree**bounded*by 2 d 2 2 + d 2 n−1 . ...
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