Irreducibility of polynomials modulo p via Newton polytopes

Gao, Shuhong; Rodrigues, Vanderlei Moraes

doi:10.1016/s0022-314x(03)00044-1

Cited by 19 publications

(19 citation statements)

References 8 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…For those cases we can gain slight improvements in separation (see Example 2). More improvements for sparse polynomials may be obtained by adapting the results in [9] which can be used to further reduce the size of R(f ) when f is sparse.…”

Section: Discussionmentioning

confidence: 99%

“…Parts of the following lemma and lemma 5 are similar to parts of the proof of Theorem 6 in [9] which are used to bound the norms of the rows of a modified Ruppert matrix. Lemma 1.…”

Section: Irreducibility Testmentioning

confidence: 99%

“…We combine an absolute irreducibility criterion by Ruppert [21,22] with the theory of the nearest rank-deficient matrix. Already Gao [8] realized the powerful algorithmic implications of Ruppert's partial differential equation, and with Rodrigues [9] applied it to absolute irreducibility of sparse polynomials modulo p. By using the original Eckart and Young theorem [5] (on complex matrices [27]) we can quickly obtain separation bounds that apply for the Euclidean distance norm on the coefficient vector of f . Our technique, applied to the example polynomial in Nagasaka's paper, produces a bound several times larger than the one he computes.…”

Section: Introductionmentioning

confidence: 99%

“…We can give an explicit formula in terms of the smallest singular value of a certain sparse matrix that has integer multiples of the coefficients of f as entries. However, like in [9] further computation can adapt to the possibly sparse structure of f and improve B(f ). In order to generalize our approach to 1-and ∞-norm, we make use of the theory of Moore-Penrose pseudo-inverses.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

On approximate irreducibility of polynomials in several variables

Kaltofen

May

2003

Proceedings of the 2003 International Symposium on Symbolic and Algebraic Computation

View full text Add to dashboard Cite

We study the problem of bounding all factorizable polynomials away from a polynomial that is absolutely irreducible. Such separation bounds are useful for testing whether a numerical polynomial is absolutely irreducible, given a certain tolerance on its coefficients. Using an absolute irreducibility criterion due to Ruppert, we are able to find useful separation bounds, in several norms, for bivariate polynomials. We also use Ruppert's criterion to derive new, more effective Noether forms for polynomials of arbitrarily many variables. These forms lead to small separation bounds for polynomials of arbitrarily many variables.

show abstract

Section: Discussionmentioning

confidence: 99%

“…Parts of the following lemma and lemma 5 are similar to parts of the proof of Theorem 6 in [9] which are used to bound the norms of the rows of a modified Ruppert matrix. Lemma 1.…”

Section: Irreducibility Testmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

On approximate irreducibility of polynomials in several variables

Kaltofen

May

2003

Proceedings of the 2003 International Symposium on Symbolic and Algebraic Computation

View full text Add to dashboard Cite

show abstract

“…In [28] The tolerance correction mechanism used in NSolve with SNAP structures assumes that the given representative polynomial does not have multiple roots. Therefore, the following warning message is generated if the given polynomial has multiple (or close) roots and any tolerance correction is not applied.…”

Section: · Examplesmentioning

confidence: 99%

Symbolic-Numeric Algebra for Polynomials

Nagasaka¹

2007

TMJ

View full text Add to dashboard Cite

Symbolic-Numeric Algebra for Polynomials (SNAP) is a prototype package that includes basic functions for computing approximate algebraic properties, such as the approximate GCD of polynomials. Our aim is to show how the unified tolerance mechanism we introduce in the package works. Using this mechanism, we can carry out approximate calculations under certified tolerances. In this article, we demonstrate the functionality of the currently released package (Version 0.2), which is downloadable from wwwmain.h.kobe-u.ac.jp/~nagasaka/research/snap/index.phtml.en. ‡ Introduction Recently, there have been many results in the area of symbolic-numeric algorithms, especially for polynomials (for example, approximate GCD for univariate polynomials [1,2,3,4] and approximate factorization for bivariate polynomials [5,6,7,8]). We think those results have practical value, which should be combined and implemented into one integrated computer algebra system such as Mathematica. In fact, Maple has such a special package called SNAP (SymbolicNumeric Algorithms for Polynomials). Currently, ours is the only such package available for Mathematica.We have been developing our SNAP package for Mathematica, which is an abbreviation for Symbolic-Numeric Algebra for Polynomials. We use algebra to mean continuous capabilities of approximate operations; for example, computing an approximate GCD between an empirical polynomial and the nearest singular polynomial computed by SNAP functions of another empirical polynomial. This continuous applicability is more important for practical computations than the number of algorithms that are already implemented, especially for average users.Our aim is to provide practical implementations of SNAP functions with a unified tolerance mechanism for polynomials like Mathematica's floating-point numbers or Kako and Sasaki's effective floating-point numbers [9]. Our idea is very simple. We only have to add new data structures representing such polynomials with tolerances and basic calculation routines and SNAP functions for the structures. At this time, only simple tolerance representations (l 2 -norm, l 1 -norm, and l ¶ -norm) for coefficient vectors of polynomials and a few SNAP functions are implemented, but the package is an ongoing project (Version 1.0 should be released in March 2007).

show abstract

Towards More Accurate Separation Bounds of Empirical Polynomials II

Nagasaka

2005

Computer Algebra in Scientific Computing

View full text Add to dashboard Cite

Faculty of 1•l uman Development, Kobe University, J apan nagasakaOma i n .h .kobe-u.ac. j p A b s t ract. VVe study the problem of bounding a polynom ial which is absolutely irreducible, away from polynomials which arc not absolutely irreducible. These separation bounds arc useful for testing whether an empirical polynomial is absolutely irreducible or not, for t he given tolera nce or error bound of its coe fficie nts. [n the former paper, we studied some improvements on I

show abstract

Irreducibility of polynomials modulo p via Newton polytopes

Cited by 19 publications

References 8 publications

On approximate irreducibility of polynomials in several variables

On approximate irreducibility of polynomials in several variables

Symbolic-Numeric Algebra for Polynomials

Towards More Accurate Separation Bounds of Empirical Polynomials II

Contact Info

Product

Resources

About