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Testing polynomials

Jean-Jaques Risler, Felice Ronga
1990 Journal of symbolic computation  
We show that any set of evaluations (a testing set) for a polynomial with k terms has to contain at least k members, and we construct such a set with precisely k members.  ...  This construction relies on knowing the exponents, but not the coefficients, of the polynomial being tested. We leave to others the task of converting this theorem into algorithms."  ... 
doi:10.1016/s0747-7171(08)80033-8 fatcat:ktu6znvqbze75ni5w6tqm45xwa

Test polynomials

Zbigniew Jelonek
2000 Journal of Pure and Applied Algebra  
We also give examples of strong test polynomial in the class of all endomorphisms of C[x1; : : : ; xn].  ...  In this paper we show that a generic polynomial p ∈ C[x 1; : : : ; xn] of degree greater than n is a strong test polynomial for monomorphisms of C[x1; : : : ; xn].  ...  If deg p ≥ 4 then a polynomial p is a test polynomial; otherwise it is not a test polynomial. Proof. First assume that a polynomial p is of degree ≥ 4. We show that p is a test polynomial.  ... 
doi:10.1016/s0022-4049(98)00135-2 fatcat:2ojcsty33vclnewo2ozgvdtbxi

Tests of Symmetric Polynomials

G. A. Miller
1911 The American mathematical monthly  
The above remarks apply directly to all rational func- tions of the n variables as well as to the more special functions which are commonly (but not universally) called polynomials.  ... 
doi:10.1080/00029890.1911.11997604 fatcat:byj65ssvzrgovej56h322nifty

Tests of Symmetric Polynomials

G. A. Miller
1911 The American mathematical monthly  
Although the last theorem requires n-1 tests to prove the symmetry of a polynomial, each of these tests is so very simple that the total number of them involve about the same amount of labor as each of  ...  TESTS OF SYMMETRIC POLYNOMIALS. By DR. G. A. MILLER, University of Illinois. * Dyck, Mathematische Annalen, Vol. 22 (1883), p. 89. t Mathematische Annalen, Vol. 33 (1889), p. 587.  ... 
doi:10.2307/2973677 fatcat:dtq3vxtsrveehlbrs4e5pxg5mm

Polynomial Time Algorithm for Graph Isomorphism Testing [article]

Michael I. Trofimov
2013 arXiv   pre-print
This article deals with new polynomial time algorithm for graph isomorphism testing.  ...  At present many effective algorithms for graph isomorphism testing were proposed and there are number of investigations with attempts to prove polynomial complexity of some of these algorithms.  ...  Now we can consider the following algorithm, where function Verify tests found mapping by procedure P1.  ... 
arXiv:1004.1808v6 fatcat:kdahgyrdijaojmizcjayac3jaa

Counting and testing dominant polynomials [article]

Artūras Dubickas, Min Sha
2015 arXiv   pre-print
In this paper, we concentrate on counting and testing dominant polynomials with integer coefficients.  ...  Finally, we will design some algorithms to test whether a given polynomial with integer coefficients is dominant or not without finding the polynomial roots.  ...  Testing dominant polynomials In this section, we will design some algorithms to test whether a given polynomial f ∈ Z[X] is dominant or not without finding the polynomial roots.  ... 
arXiv:1407.2789v2 fatcat:ibp2lbwy6zavfb4j5ybvtarluu

Testing hyperbolicity of real polynomials [article]

Papri Dey, Daniel Plaumann
2018 arXiv   pre-print
Hyperbolic polynomials are real multivariate polynomials with only real roots along a fixed pencil of lines. Testing whether a given polynomial is hyperbolic is a difficult task in general.  ...  We examine different ways of translating hyperbolicity into nonnegativity conditions, which can then be tested via sum-of-squares relaxations.  ...  Finally, it should also be pointed out that we always test hyperbolicity of a polynomial with respect to a fixed point e.  ... 
arXiv:1810.04055v2 fatcat:4qcuvrjntrdvfae3mnccpnu5sm

Test complexity of generic polynomials

Peter Bürgisser, Thomas Lickteig, Michael Shub
1992 Journal of Complexity  
INTRODUCTION Given a polynomial f: Cm + C we may check whether f(t) = 0 by evaluating fat 4. But testing for zero may be easier than evaluating.  ...  0) such that for any polynomial f E k[x, , . . . , x,] of degree d ~*.k[xl(f) 5 j& (1 + Pd).  ... 
doi:10.1016/0885-064x(92)90022-4 fatcat:3j3xkv37ofawzeujun2uuej4lm

Read-once polynomial identity testing

Amir Shpilka, Ilya Volkovich
2008 Proceedings of the fourtieth annual ACM symposium on Theory of computing - STOC 08  
In this paper we study the problems of giving deterministic identity testing for models related to preprocessed ROFs.  ...  A preprocessed ROF (PROF for short) is a ROF in which we are allowed to replace each variable x i with a univariate polynomial T i (x i ).  ...  Introduction In this paper we study the polynomial identity testing problem for several models based on readonce formulas.  ... 
doi:10.1145/1374376.1374448 dblp:conf/stoc/ShpilkaV08 fatcat:h2nph4eyurfflnlpdcckgkfx3e

Read-once polynomial identity testing

Amir Shpilka, Ilya Volkovich
2015 Computational Complexity  
In this paper we study the problems of giving deterministic identity testing for models related to preprocessed ROFs.  ...  A preprocessed ROF (PROF for short) is a ROF in which we are allowed to replace each variable x i with a univariate polynomial T i (x i ).  ...  Introduction In this paper we study the polynomial identity testing problem for several models based on readonce formulas.  ... 
doi:10.1007/s00037-015-0105-8 fatcat:qfbq4xbyd5fdfdh2hsfryzeuou

Likelihood ratio tests for positivity in polynomial regressions [article]

Naohiro Kato, Satoshi Kuriki
2012 arXiv   pre-print
In this paper, we consider the likelihood ratio test for the hypothesis of positivity that the estimand polynomial regression curve is a positive polynomial.  ...  A polynomial that is nonnegative over a given interval is called a positive polynomial. The set of such positive polynomials forms a closed convex cone K.  ...  likelihood ratio test (LRT) for testing H 1 against H 2 .  ... 
arXiv:1108.1033v4 fatcat:3tzaxi7fcvdurcqr27s3lrozxq

Polynomial Identity Testing for Low Degree Polynomials with Optimal Randomness

Markus Bläser, Anurag Pandey, Raghu Meka, Jarosław Byrka
2020 International Workshop on Approximation Algorithms for Combinatorial Optimization  
We give a randomized polynomial time algorithm for polynomial identity testing for the class of n-variate poynomials of degree bounded by d over a field 𝔽, in the blackbox setting.  ...  We also collect two simple constructions of hitting sets with information theoretically optimal size against the class of n-variate, degree d polynomials.  ...  For a history on the progress on polynomial identity testing, we refer the readers to [47, 42, 43] . In this work, we are interested in blackbox polynomial identity testing.  ... 
doi:10.4230/lipics.approx/random.2020.8 dblp:conf/approx/BlaserP20 fatcat:soynqhsaqzgt5peas7wr52y7ia

T1 testing implies Tp polynomial testing: optimal cancellation conditions for CZO's [article]

Eric T. Sawyer
2019 arXiv   pre-print
The main result here is that the familiar T1 testing conditions over indicators of cubes, together with the one-tailed A2 conditions, imply polynomial testing.  ...  Controlling polynomial testing conditions -main theorems.  ...  Introduction and definitions In Theorem 3 below, we show that for fractional Calderón-Zygmund operators, the κ-Cube Testing conditions over polynomials of degree less than κ times indicators of cubes,  ... 
arXiv:1907.10734v2 fatcat:5kbliabv7rbadl6i3c6g6irqcy

Equivalence of Polynomial Identity Testing and Deterministic Multivariate Polynomial Factorization

Swastik Kopparty, Shubhangi Saraf, Amir Shpilka
2014 2014 IEEE 29th Conference on Computational Complexity (CCC)  
In this paper we show that the problem of deterministically factoring multivariate polynomials reduces to the problem of deterministic polynomial identity testing.  ...  of f can be solved deterministically, given a deterministic algorithm for the polynomial identity testing problem (we require either a white-box or a black-box algorithm, depending on the representation  ...  Claim 3.5 The pair g k , h k obtained above is the unique pair of polynomials (mod T 2 k ) such that (a) f = g k h k mod T 2 k , (b) g k is monic in X, and (c) g k = g 0 mod T .  ... 
doi:10.1109/ccc.2014.25 dblp:conf/coco/KoppartySS14 fatcat:crhpkkfubvhitdpfn3jc56jdwi

On testing the divisibility of lacunary polynomials by cyclotomic polynomials

Michael Filaseta, Andrzej Schinzel
2003 Mathematics of Computation  
A running analysis shows that, for a fixed number of nonzero terms, the algorithm runs in polynomial time.  ...  The algorithm is intended to be used for sparse polynomials given as a sequence of coefficientexponent pairs.  ...  THE DIVISIBILITY OF LACUNARY POLYNOMIALS  ... 
doi:10.1090/s0025-5718-03-01589-8 fatcat:6amujzwlbrhqhgbrqqjjqsaufa
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