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On Polynomial Time Computable Numbers
[article]

2006
*
arXiv
*
pre-print

It will be shown that the

arXiv:cs/0608067v1
fatcat:xjpavj7rofaqvbsbtnc2rugeba
*polynomial**time**computable**numbers*form a field, and especially an algebraically closed field. ...*Computable**number*means that it can be approximated by a*computable*function, and*polynomial**time**computable**number*by a*polynomial**time**computable*function. ... The Field of*Polynomial**Time**Computable**Numbers*In this section, we show that the whole*polynomial**time**computable**numbers*C P is a field. ...##
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Polynomial time quantum algorithm for the computation of the unit group of a number field

2005
*
Proceedings of the thirty-seventh annual ACM symposium on Theory of computing - STOC '05
*

This algorithm is applied to

doi:10.1145/1060590.1060661
dblp:conf/stoc/SchmidtV05
fatcat:t4zx37j7tjhzrgdkh7dugkhllm
*compute*the unit group of finite extensions of Q. Execution*time*for fixed field degree over Q is*polynomial*in the discriminant of the field. ... We present a quantum algorithm for the*computation*of the irrational period lattice of a function*on*Z n which is periodic in a relaxed sense. ... His algorithm executes in*polynomial**time*provided function values can be*computed*within*time**polynomial*in the period and the size of the arguments. ...##
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Market Equilibria in Polynomial Time for Fixed Number of Goods or Agents

2008
*
2008 49th Annual IEEE Symposium on Foundations of Computer Science
*

A final note

doi:10.1109/focs.2008.30
dblp:conf/focs/DevanurK08
fatcat:er6ei5fxxbcdfonwrz7laqqn3q
*on**computing*approximate 4 equilibria: for constant*number*of goods, there are various algorithms that*compute*an approximate equilibria in*time*exponential in the*number*of goods, such as ... equilibrium in*polynomial**time*. ... Running*Time*: If m is a constant, then the*number*of*polynomials*and their degree in each step are*polynomials*. And the*number*of variables is a constant. Hence the running*time*is*polynomial*. ...##
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A polynomial time algorithm for computing the HNF of a module over the integers of a number field

2012
*
Proceedings of the 37th International Symposium on Symbolic and Algebraic Computation - ISSAC '12
*

The modular strategy was conjectured to run in

doi:10.1145/2442829.2442844
dblp:conf/issac/BiasseF12
fatcat:e6kym36d35h2fgxn6kpp5finby
*polynomial**time*by Cohen, but so far, no such proof was available in the literature. ... We present a variation of the modular algorithm for*computing*the Hermite Normal Form of an OK -module presented by Cohen [2], where OK is the ring of integers of a*number*field K. ... This result is significant since other applications rely*on*the possibility of*computing*the HNF of an OK -module in*polynomial**time*. ...##
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A polynomial-time algorithm to approximately count contingency tables when the number of rows is constant

2002
*
Proceedings of the thiry-fourth annual ACM symposium on Theory of computing - STOC '02
*

In this case, we show that the solution lies in approximating the volume of a single convex body, a problem which is known to be solvable in

doi:10.1145/509907.509946
dblp:conf/stoc/CryanD02
fatcat:d7mfmzwftfgnfnk3nwk3raurju
*polynomial**time*[5] . ... We show that the*number*of contingency tables can be expressed as the weighted sum of a*polynomial**number*of new instances of the problem, where each instance consists of some new row sums and the original ... By dynamic programming, we can count the*number*of contingency tables*on*the small columns for any given list of partial row sums in*polynomial**time*. ...##
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A Polynomial Time Presburger Criterion and Synthesis for Number Decision Diagrams

*
20th Annual IEEE Symposium on Logic in Computer Science (LICS' 05)
*

In this paper, we show how to determine in

doi:10.1109/lics.2005.2
dblp:conf/lics/Leroux05
fatcat:euvwnjx64ffdblzvvfk6wjhqpm
*polynomial**time*whether a NDD represents a Presburger-definable set, and we provide in this positive case a*polynomial**time*algorithm that constructs from the ...*Number*Decision Diagrams (NDD) are the automatabased symbolic representation for manipulating sets of integer vectors encoded as strings of digit vectors (least or most significant digit first). ... Acknowledgment: We thank Pierre McKenzie for his support and for his interesting remarks*on*so many versions of this paper. ...##
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A polynomial-time algorithm for the perfect phylogeny problem when the number of character states is fixed

*
Proceedings of 1993 IEEE 34th Annual Foundations of Computer Science
*

C1) S c V ( T ) c A1 x . . . x A" W e present a

doi:10.1109/sfcs.1993.366873
dblp:conf/focs/AgarwalaF93
fatcat:tmrkc7pevvfmvaajvpoeift33i
*polynomial*-*time*algorithm f o r determining whether a set of species, described by the characters they exhibit, has a perfect phylogeny, assuming the maximum ...*number*of possible states f o r a charact e r is fixed. ... Checking whether a split (GI, G2) is of type I can be achieved O(nm)*time*as follows. First,*compute*M(G1) and the common state between G1 and G2*on*each c E M(G1). ...##
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Computable real function F such that F is not polynomial time computable on [0,1]
[article]

2014
*
arXiv
*
pre-print

A

arXiv:1404.7053v1
fatcat:dr5axojtu5akjir2usxolkpkvm
*computable*real function F*on*[0,1] is constructed such that there exists an exponential*time*algorithm for the evaluation of the function*on*[0,1]*on*Turing machine but there does not exist any*polynomial*...*time*algorithm for the evaluation of the function*on*[0,1]*on*Turing machine (moreover, it holds for any rational point*on*(0,1)) ... Theorem 3 . 3 Real function F is not a*polynomial**time**computable*real function*on*[0, 1] of real*numbers*. ...##
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Computational complexity of real functions

1982
*
Theoretical Computer Science
*

Recursive analysis, the theory of

doi:10.1016/s0304-3975(82)80003-0
fatcat:njl6vcb7kjhx3owlluzvmis4ei
*computation*of functions*on*real*numbers*, has been studied from various aspects. ... We study the complexity of the roots and the differentiability of*polynomial**time**computable*real functions. ... We will see later that all algebraic*numbers*are*polynomial**time**computable*. e and 'IT are*polynomial**time**computable*. However, not every recursive real*number*is*polynomial**time**computable*. ...##
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Polynomial running times for polynomial-time oracle machines
[article]

2017
*
arXiv
*
pre-print

Thereby making it possible to consider functions

arXiv:1704.01405v2
fatcat:heacap7njrbjbey7otxmrwij34
*on*the natural*numbers*as running*times*of oracle Turing machines and avoiding second-order*polynomials*, which are notoriously difficult to handle. ... The new notion is named "strong*polynomial*-*time**computability*", and proven to be a strictly stronger requirement than*polynomial*-*time**computability*. ...*polynomial*-*time**computability*of a functional*on*Σ * * means. ...##
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Computing power series in polynomial time

1988
*
Advances in Applied Mathematics
*

It is shown that if a real-valued function f is

doi:10.1016/0196-8858(88)90006-1
fatcat:ic26g2wldng4fbebjwkajxzgd4
*polynomial*-*time**computable**on*[a, b], with a < 0 < b, and is analytic at 0, then the Taylor coefficients off at 0, as a sequence of real*numbers*, is*polynomial*-*time*...*computable*. 0 19x8 Academic PESS. ... Our main result shows that the derivatives of a*polynomial*-*time**computable*, analytic function form a*polynomial*-*time**computable*sequence of real*numbers*. ...##
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New non-uniform segmentation technique for software function evaluation

2016
*
2016 IEEE 27th International Conference on Application-specific Systems, Architectures and Processors (ASAP)
*

Indeed,

doi:10.1109/asap.2016.7760782
dblp:conf/asap/BonnotNM16
fatcat:7z7aiimbbfbvbl76zw2ixie2ae
*polynomial*approximation methods allow to find a trade-off between accuracy and*computation**time*. Software implementation of*polynomial*approximation in fixed-point processors is considered. ... This paper presents a method to*compute*the values of a function*on*I using non-uniform segmentation, and*polynomial*approximation. ... be*computed*or the*computation**time*, knowing the size of the indexing table T and of the coefficients table P as well as the equation giving the*computation**time*depending*on*the degree N l and the*number*...##
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On the complexity of finding circumscribed rectangles and squares for a two-dimensional domain

2006
*
Journal of Complexity
*

We study this problem in the

doi:10.1016/j.jco.2006.05.005
fatcat:g2m3rioevffavo5kemkrrykgbi
*polynomial*-*time*complexity theory of real functions based*on*the oracle Turing machine model. ... We show that for any*polynomial*-*time**computable*Jordan curve , there must exist at least*one**computable*circumscribed square (not necessarily of the minimum area), but this square may have arbitrarily ... We say a rectangle R is*polynomial*-*time**computable*if its four corners are*polynomial*-*time**computable*complex*numbers*. ...##
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Some negative results on the computational complexity of total variation and differentiation

1982
*
Information and Control
*

There exists a

doi:10.1016/s0019-9958(82)91083-x
fatcat:7lpmmgf5ofhdvly66lsy46vhfu
*polynomial**time**computable*function f*on*[0, 1] such that f' exists and is continuous*on*[0, 1] burr' is not*polynomial**time**computable*. ... A real*number*x is*polynomial**time**computable*if there is a sequence {dn} of dyadic rationals binary converging to x and the function 20"[d,] is*polynomial**time**computable*. ... Note added in proof The author has obtained the following stronger version of Theorem4: there exists a*polynomial**time**computable*function f*on*[0, 1] having a continuous derivativef'*on*[0, 1] such that ...##
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Computing a Solution of Feigenbaum's Functional Equation in Polynomial Time

2014
*
Logical Methods in Computer Science
*

We show that this solution is a

doi:10.2168/lmcs-10(4:7)2014
fatcat:kmkas7f7izatrmgse2hqa5mt2e
*polynomial**time**computable*function. This implies in particular that the so-called first Feigenbaum constant is a*polynomial**time**computable*real*number*. ... The proof is based*on*a*number*of claims in Lanford's paper [10] . ... We intend to see how many digits of the first Feigenbaum constant we can*compute*with a correctness guarantee, using the exact real*number*arithmetic package iRRAM by Müller [12] . ...
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