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Formal Proofs of Transcendence for e and π as an Application of Multivariate and Symmetric Polynomials [article]

Sophie Bernard, Laurence Rideau, Pierre-Yves Strub
2015 arXiv   pre-print
The case of π relies extensively on properties of multivariate polynomials and this experiment was also an occasion to put to test a newly developed library for these multivariate polynomials.  ...  We describe the formalisation in Coq of a proof that the numbers e and π are transcendental.  ...  The more complex proof of transcendence for π was not formalized yet.  ...

Formalization of the Lindemann-Weierstrass Theorem [chapter]

Sophie Bernard
2017 Lecture Notes in Computer Science
As we follow Baker's proof, we discuss the difficulties of its formalization and explain how we resolved them in Coq.  ...  Most of these difficulties revolve around multivariate polynomials and their relationship with the conjugates of a univariate polynomial.  ...  As the context for this proof is almost exactly the same as for the direct proof of e and π, you can refer to  for more details.  ...

Series with general exponents [article]

Daniel E. Loeb
1995 arXiv   pre-print
As an application, we compute the algebra of symmetric functions with nonnegative real exponents. The applications to logarithmic series and the Umbral calculus are deferred to another paper.  ...  We define the Artinian and Noetherian algebra which consist of formal series involving exponents which are not necessarily integers. All of the usual operations are defined here and characterized.  ...  For these applications, it must be proven that one can manipulate formal power series with real exponents as easily as one does polynomials.  ...

On the Non-Holonomic Character of Logarithms, Powers, and the \$n\$th Prime Function

Philippe Flajolet, Stefan Gerhold, Bruno Salvy
2005 Electronic Journal of Combinatorics
Our proofs depend on basic complex analysis, namely a conjunction of the Structure Theorem for singularities of solutions to linear differential equations and of an Abelian theorem.  ...  We establish that the sequences formed by logarithms and by "fractional" powers of integers, as well as the sequence of prime numbers, are non-holonomic, thereby answering three open problems of Gerhold  ...  Thanks to an anonymous referee for his supportive assessment of our article.  ...

Painlevé Transcendent Evaluations of Finite System Density Matrices for 1d Impenetrable Bosons

P.J. Forrester, N.E. Frankel, T.M. Garoni, N.S. Witte
2003 Communications in Mathematical Physics
For the impenetrable Bose gas in a harmonic trap, and with Dirichlet or Neumann boundary conditions, we give a determinant form for the density matrix, a form as an average over the eigenvalues of an ensemble  ...  of random matrices, and in special cases an evaluation in terms of a transcendent related to Painlevé V and VI.  ...  Korepin and A. Fetter for comments and suggestions on this work.  ...

Distributed Algorithmic Mechanism Design and Algebraic Communication Complexity [chapter]

Markus Bläser, Elias Vicari
2008 Lecture Notes in Computer Science
value of a polynomial or rational function depending on an input distributed between the two players.  ...  We define a general algebraic model over an arbitrary field k of characteristic 0, where the involved functions can be computed with the natural operations additions, multiplications and divisions and  ...  Shankar Ram and Andreas Meyer for carefully reading an early draft of this paper and in particular the latter for the help provided in proving Lemma 1.  ...

Positive polynomials in scalar and matrix variables, the spectral theorem and optimization [article]

John William Helton, Mihai Putinar
2006 arXiv   pre-print
These new applications have prompted a series of recent studies devoted to the structure of positivity and convexity in a free *-algebra, the appropriate setting for analyzing inequalities on polynomials  ...  The second part of the survey focuses on recently discovered connections between real algebraic geometry and optimization as well as polynomials in matrix variables and some control theory problems.  ...  It is clear that |p(e it )| = |p ♭ (e it )|, t ∈ [−π, π], and that the roots of p ♭ are symmetric with respect to the unit circle to the roots of p.  ...

Parametric Model Checking Continuous-Time Markov Chains

Catalin-Andrei Ilie, James Ben Worrell, Martin Theobald, Ana Ozaki, Emilio Muñoz-Velasco
2020 International Symposium/Workshop on Temporal Representation and Reasoning
A second contribution is to give a reduction of the Positivity Problem for matrix exponentials to the PCSL model checking problem, suggesting that it will be difficult to give an unconditional proof of  ...  CSL is a well-known temporal logic for specifying properties of real-time stochastic systems, such as continuous-time Markov chains.  ...  References A Proof of Thereom 4 Let f (t) = m j=1 P j (t)e λj t , together with the interval [c, d] be an instance of the Positivity Problem for exponential polynomials.  ...

Hypergeometric Expressions for Generating Functions of Walks with Small Steps in the Quarter Plane [article]

Alin Bostan, Frédéric Chyzak, Mark van Hoeij, Manuel Kauers, Lucien Pech
2016 arXiv   pre-print
We concern ourselves with the enumeration of such walks starting at the origin and constrained to remain in the quarter plane N^2, counted by their length and by the position of their ending point.  ...  As a first corollary, we prove that all these 19 generating functions can be expressed in terms of Gauss' hypergeometric functions that are intimately related to elliptic integrals.  ...  In other words, (e 1 − i ′ + C 1 ) ∩ (e 2 − i ′ + C 2 ) ∩ π −1 2 (0) is bounded, and, after shifting by i ′ , so is the set (e 1 + C 1 ) ∩ (e 2 + C 2 ) ∩ π −1 2 (i).  ...

Hypergeometric expressions for generating functions of walks with small steps in the quarter plane

Alin Bostan, Frédéric Chyzak, Mark van Hoeij, Manuel Kauers, Lucien Pech
2017 European journal of combinatorics (Print)
., that satisfies a linear differential equation with polynomial coefficients. The step set S = {(1, 1), (−1, 1), (0, −1)} is an example for this case [9, 12] .  ...  ., it satisfies a polynomial equation P (t, Q(t)) = 0 for some P ∈ Q[t, T ] \ {0}. But this is not the case for all other step sets.  ...  In other words, (e 1 − i + C 1 ) ∩ (e 2 − i + C 2 ) ∩ π −1 2 (0) is bounded, and, after shifting by i , so is the set (e 1 + C 1 ) ∩ (e 2 + C 2 ) ∩ π −1 2 (i).  ...

COMPUTABLE IMPLEMENTATION OF ``FUNDAMENTAL THEOREM OF ALGEBRA"

J.A. Sjogren, X. Li, M. Zhao, C. Lu
2013 International Journal of Pure and Applied Mathematics
Some examples show how these routines apply to the algebra of symmetric multinomial forms used in Laplace's proof (1795) of FTA, as well as to the theory of Sylvester forms and the Bézoutian formulation  ...  The Fundamental Theorem of Algebra (FTA) has been studied for more than 300 years: more or less satisfactory proofs of FTA emerged in the 18th and 19th centuries.  ...  Acknowledgments This research project has been supported by the Air Force Office of Scientific Research, FA9550-11-1-0315.  ...

Zeroes and rational points of analytic functions [article]

Georges Comte, Yosef Yomdin
2017 arXiv   pre-print
the graph Γ_f of f_ D and algebraic curves of degree d is polynomially bounded in d.  ...  As a consequence, for any function f in these families, Γ_f has less than β^α T rational points of height at most T, for some α, β >0.  ...  The authors would like to thank University of Savoie Mont Blanc and the Weizmann Institute for the hospitality during part of the research for this paper.  ...

Zeroes and rational points of analytic functions

Georges Comte, Yosef Yomdin
2018 Annales de l'Institut Fourier
-For f as above and for ∆ the absolute value of a non-zero minor determinant of size (3.3) are satisfied. Then on D 1 4 Z d (f ) T (d),for a certain polynomial T .Proof.  ...  For a polynomial Q of arity m and for f = (a k ) k 0 ∈ I ∞ we denote Q(a 0 , . . . , a m−1 ) = Q(π m (f )) by Q(f ).  ...

Counting solutions of differential equations

Markus Lange-Hegermann
2015 ACM Communications in Computer Algebra
Consider a multivariate polynomial p ∈ R as the family of univariate polynomials φ <ld(p),a (p) for certain a ∈ F n .  ...  Thus, every time an initial is added to the system as an inequation, divide the polynomial by its content. Additionally, the multivariate content, which is an element of the field F , can be removed.  ...  Kolchin in [Kol73, §IV.9] The proof of the differential dimension polynomial Section 1.7 includes statements about prime decompositions of radical algebraic and differential ideals.  ...

On the Transcendence of Period Images [article]

David Urbanik
2021 arXiv   pre-print
As a corollary we establish the existence of a canonical ℚ-algebraic model for normalizations of period images.  ...  We introduce new differential-algebraic techniques to show this is true for all points s ∈ S(ℚ) outside of an explicit proper closed algebraic subset of S.  ...  To compute the coefficient cp of the t term in Equation 1, for instance, we may use several applications of the multivariate chain rule so that the constant term of the resulting formal expression is a  ...
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