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A characterization of the 0-basis homogeneous bounding degrees

Karen Lange
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 .  ... 
doi:10.2178/jsl/1278682211 fatcat:nw6krpdwsfdull2olyp4agacmu

The degree spectra of homogeneous models

Karen Lange
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.  ... 
doi:10.2178/jsl/1230396762 fatcat:il3q54zgozajtfsgunepxnnnc4

Computability of Homogeneous Models

Karen Lange, Robert I. Soare
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.  ... 
doi:10.1305/ndjfl/1172787551 fatcat:bbfqknx62jcfhgkhmsjfqrn2ty

On the complexity of computing Gröbner bases for weighted homogeneous systems

Jean-Charles Faugère, Mohab Safey El Din, Thibaut Verron
2016 Journal of symbolic computation  
Furthermore, 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  ... 
doi:10.1016/j.jsc.2015.12.001 fatcat:5xnhv5awu5dttgraorrc7dvo7m

Bases in spaces of homogeneous polynomials on Banach spaces

Leonardo Pellegrini
2007 Journal of Mathematical Analysis and Applications  
In this work we present some conditions 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.  ... 
doi:10.1016/j.jmaa.2006.09.058 fatcat:j53bp5byxjenvffhfgcrybcwju

On the Universal Gröbner bases of varieties of minimal degree

Sonja Petrovic
2008 Mathematical Research Letters  
This leads to 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.  ... 
doi:10.4310/mrl.2008.v15.n6.a11 fatcat:mgbe7y6yvrf33kcjhtiq2tobn4

Computing Minimal Generating Sets of Invariant Rings of Permutation Groups with SAGBI-Gröbner Basis

Nicolas Thiéry
2001 Discrete Mathematics & Theoretical Computer Science  
Our algorithm does not require 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.  ... 
doi:10.46298/dmtcs.2285 fatcat:zrgiviiqyfclvpthedwsl5aame

On the universal Gröbner bases of varieties of minimal degree [article]

Sonja Petrović
2007 arXiv   pre-print
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 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.  ... 
arXiv:0711.2714v2 fatcat:5m7owwmudjdfzcmefnvsbqms5u

Anisotropic Two-Microlocal Spaces and Regularity

Mourad Ben Slimane
2014 Journal of Function Spaces  
We prove that these spaces allow 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.  ... 
doi:10.1155/2014/505796 fatcat:s3wmp4agifge5ciaklqmnrio5y

Homogenization and the Polynomial Calculus [chapter]

Josh Buresh-Oppenheim, Toniann Pitassi, Matt Clegg, Russell Impagliazzo
2000 Lecture Notes in Computer Science  
In standard implementations 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.  ... 
doi:10.1007/3-540-45022-x_78 fatcat:bvzx63ozmfar3fjhd6ykbjrhwq

Homogenization and the polynomial calculus

Joshua Buresh-Oppenheim, Matthew Clegg, Russell Impagliazzo, Toniann Pitassi
2002 Computational Complexity  
In standard implementations 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.  ... 
doi:10.1007/s00037-002-0171-6 fatcat:4wedhdl5obemtbrvwv2oosirzm

Partition Theorems and Computability Theory

Joseph R. Mileti
2005 Bulletin of Symbolic Logic  
We denote 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.  ... 
doi:10.2178/bsl/1122038995 fatcat:auevbp5frjgsplkxupnaqpbr6e

The Degree of Regularity of HFE Systems [chapter]

Vivien Dubois, Nicolas Gama
2010 Lecture Notes in Computer Science  
Using this property with 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 .  ... 
doi:10.1007/978-3-642-17373-8_32 fatcat:a36u2tmygvfttiz2dykikxbk7q

On polynomial ideals, their complexity, and applications [chapter]

Ernst W. Mayr
1995 Lecture Notes in Computer Science  
We discuss complexity results known for 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 ?  ... 
doi:10.1007/3-540-60249-6_42 fatcat:soxs7hun4feqddpldpv2razz5y

Some Complexity Results for Polynomial Ideals

Ernst W. Mayr
1997 Journal of Complexity  
In this paper, we survey some 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 .  ... 
doi:10.1006/jcom.1997.0447 fatcat:d3cz6pfw3za4disutcnitfvlsa
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