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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.  ...  [alpha i] * + c^(n * p n k gamma c alpha))%R \in Cint Instantiating the common lemma: cases of e and pi The proof of transcendence of e is a very simple instantiation of the common lemma and the formal  ... 
arXiv:1512.02791v1 fatcat:s4hjz6yryjgaxkqi22sxkgatve

Series with general exponents

Daniel E Loeb
1991 Journal of Mathematical Analysis and Applications  
For these applications, it must be proven that one can manipulate formal power series with real exponents as easily as one does polynomials.  ...  By the above reasoning, the elementary and complete symmetric functions e,(x) and h,(x) each form an Artinian transcendence basis for /1*, and a Noetherian transcendence basis for A.  ... 
doi:10.1016/0022-247x(91)90390-l fatcat:fryvwzqw3zhzxaws2rxwiypoi4

Subtyping arithmetical types

Joseph (Yossi) Gil
2001 Proceedings of the 28th ACM SIGPLAN-SIGACT symposium on Principles of programming languages - POPL '01  
We offer a compact representation of the types in this system as multivariate algebraic functions.  ...  Mockingbird is a prototype tool for developing inter-language and distributed applications.  ...  Acknowledgments The comments of Jens Palsberg on a preliminary version of this paper are gratefully acknowledged.  ... 
doi:10.1145/360204.360232 dblp:conf/popl/Gil01 fatcat:z2watuflkbaa7ombwanufsmmgi

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.  ...  Likewise for a polynomial p ∈ R x , the Hessian p ′′ (x)[h] of p(x) can be thought of as the formal second directional derivative of p in the "direction" h.  ... 
arXiv:math/0612103v1 fatcat:4molh5aq2nfotbbpw7j3h3w6yq

Zero-equivalence in function fields defined by algebraic differential equations

John Shackell
1993 Transactions of the American Mathematical Society  
We consider function fields obtained as towers over the field of rational functions, each extension being by a solution of an algebraic differential equation.  ...  On the assumption that an oracle exists for the constants, we present two algorithms for determining whether a given expression is functionally equivalent to zero in such a field.  ...  , be the homomorphism generated by substituting /j and its derivatives for the formal parameters y,-,y¿, ... , y}"' . Then Pi : O, -» Ä, is defined as p¡ = u¡ o p¡.  ... 
doi:10.1090/s0002-9947-1993-1088022-2 fatcat:g2p7ndag3van5i6m4u7nhqb32y

Zero-Equivalence in Function Fields Defined by Algebraic Differential Equations

John Shackell
1993 Transactions of the American Mathematical Society  
We consider function fields obtained as towers over the field of rational functions, each extension being by a solution of an algebraic differential equation.  ...  On the assumption that an oracle exists for the constants, we present two algorithms for determining whether a given expression is functionally equivalent to zero in such a field.  ...  , be the homomorphism generated by substituting /j and its derivatives for the formal parameters y,-,y¿, ... , y}"' . Then Pi : O, -» Ä, is defined as p¡ = u¡ o p¡.  ... 
doi:10.2307/2154342 fatcat:7l7y6p6z2jcunc2ar32kb2yuy4

Diagonalization and Rationalization of algebraic Laurent series [article]

Boris Adamczewski, Jason P. Bell
2012 arXiv   pre-print
As a consequence, we obtain that for every prime p the reduction modulo p of the diagonal of a multivariate algebraic power series f with integer coefficients is an algebraic power series of degree at  ...  most p^A and height at most A^2p^A+1, where A is an effective constant that only depends on the number of variables, the degree of f and the height of f.  ...  For instance the polynomial x 2 y 3 ∈ K[x, y] has degree 5 but viewed as an element of K[[x, y]] it is an algebraic power series of degree 1.  ... 
arXiv:1205.4090v1 fatcat:smlmhbnkc5a73hcpwrvah5f7ha

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.  ... 
doi:10.1145/2768577.2768622 fatcat:awufukj6xrac5gztufh6yikp6q

Two-variable polynomials with dynamical Mahler measure zero [article]

Annie Carter, Matilde Lalín, Michelle Manes, Alison Beth Miller, Lucia Mocz
2021 arXiv   pre-print
We discuss several aspects of the dynamical Mahler measure for multivariate polynomials.  ...  We prove a weak dynamical version of Boyd--Lawton formula and we characterize the polynomials with integer coefficients having dynamical Mahler measure zero both for the case of one variable (Kronecker's  ...  Mahler measure for multivariate polynomials was first considered by Mahler [Mah62] in connection to heights and their applications in transcendence theory.  ... 
arXiv:2110.06496v1 fatcat:hv5hyhudczd4zpd3rrlquwjfim

Random matrix theory

Alan Edelman, N. Raj Rao
2005 Acta Numerica  
Especially, we thank Arieh Iserles and Brad Baxter for their comments and Random matrix theory 59 encouragement, Glennis Starling for being incredibly patient with us in the face of severe time constraints  ...  In particular, portions of Ioana's (Dumitriu 2003 ) and Brian's (Sutton 2005 ) dissertation work formed the basis for Section 4 and Section 11 respectively.  ...  We can create an n × n symmetric matrix A by, for example, creating an n × n matrix X with independent Gaussian entries and then symmetrizing it as A = (X + X )/n.  ... 
doi:10.1017/s0962492904000236 fatcat:pccnr3o4xbdk3isudg3t2dwsya

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.  ... 
doi:10.4230/lipics.time.2020.7 dblp:conf/time/IlieW20 fatcat:srnjorcsqzgila53crnjxfp5tm

Caustics, counting maps and semi-classical asymptotics [article]

N. M. Ercolani
2010 arXiv   pre-print
The coefficients of the large N expansion of the logarithm of this partition function,also known as the genus expansion, (and its derivatives) are generating functions for a variety of graphical enumeration  ...  These results in turn provide new information about the asymptotics of recurrence coefficients for orthogonal polynomials with respect to exponential weights.  ...  m λ (x 1 , . . . , x ν+1 ) is the monomial symmetric polynomial associated to λ [21].  ... 
arXiv:0912.1904v2 fatcat:2eepsdzmjrgk3jg5crrloyu3qu

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  ... 
arXiv:2106.09342v1 fatcat:6a65zpaapnbo7kj64yviyjodnq

Real algebraic geometry

2014 ChoiceReviews  
We also wish to thank the numerous sponsors who allowed us to welcome so many participants under proper conditions, in particular a large number of PhD students and post-PhD students called to constitute  ...  the pool of the Real Algebraic Geometry of tomorrow.  ...  An explicit expression of f as a sum of squares is a certificate of positivity for f , i.e., a polynomial identity which gives an immediate proof of the positivity of of f on R n .  ... 
doi:10.5860/choice.51-3296 fatcat:xm5l6ieafzewtfpuegkxsbtzhy

The summer meeting in ann Arbor

J. W. T. Youngs
1955 Bulletin of the American Mathematical Society  
The present paper gives an abstract definition for each group in terms of two, three, or four generators.  ...  ., the 17 ways of repeating a flat design, as on wallpaper) were discovered empirically by the Moors in their decoration of the Alhambra.  ...  Let P be a partially ordered set with elements pi and order relation (^), and let the set, CF(P) f of lattice polynomials (words) be ordered by: (i) pi^pj if and only if pi^pj in P, and, (ii) recursively  ... 
doi:10.1090/s0002-9904-1955-09979-8 fatcat:4rmgfntcavcclkvmixl6xgqg34
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