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

2015
*
arXiv
*
pre-print

The case

arXiv:1512.02791v1
fatcat:s4hjz6yryjgaxkqi22sxkgatve
*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*...##
###
Series with general exponents

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. ...

##
###
Subtyping arithmetical types

2001
*
Proceedings of the 28th ACM SIGPLAN-SIGACT symposium on Principles of programming languages - POPL '01
*

We offer a compact representation

doi:10.1145/360204.360232
dblp:conf/popl/Gil01
fatcat:z2watuflkbaa7ombwanufsmmgi
*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. ...##
###
Positive polynomials in scalar and matrix variables, the spectral theorem and optimization
[article]

2006
*
arXiv
*
pre-print

These new

arXiv:math/0612103v1
fatcat:4molh5aq2nfotbbpw7j3h3w6yq
*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. ...##
###
Zero-equivalence in function fields defined by algebraic differential equations

1993
*
Transactions of the American Mathematical Society
*

We consider function fields obtained

doi:10.1090/s0002-9947-1993-1088022-2
fatcat:g2p7ndag3van5i6m4u7nhqb32y
*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¡. ...##
###
Zero-Equivalence in Function Fields Defined by Algebraic Differential Equations

1993
*
Transactions of the American Mathematical Society
*

We consider function fields obtained

doi:10.2307/2154342
fatcat:7l7y6p6z2jcunc2ar32kb2yuy4
*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¡. ...##
###
Diagonalization and Rationalization of algebraic Laurent series
[article]

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. ...

##
###
Counting solutions of differential equations

2015
*
ACM Communications in Computer Algebra
*

Consider a

doi:10.1145/2768577.2768622
fatcat:awufukj6xrac5gztufh6yikp6q
*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. ...##
###
Two-variable polynomials with dynamical Mahler measure zero
[article]

2021
*
arXiv
*
pre-print

We discuss several aspects

arXiv:2110.06496v1
fatcat:hv5hyhudczd4zpd3rrlquwjfim
*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. ...##
###
Random matrix theory

2005
*
Acta Numerica
*

Especially, we thank Arieh Iserles

doi:10.1017/s0962492904000236
fatcat:pccnr3o4xbdk3isudg3t2dwsya
*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. ...##
###
Parametric Model Checking Continuous-Time Markov Chains

2020
*
International Symposium/Workshop on Temporal Representation and Reasoning
*

A second contribution is to give a reduction

doi:10.4230/lipics.time.2020.7
dblp:conf/time/IlieW20
fatcat:srnjorcsqzgila53crnjxfp5tm
*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*. ...##
###
Caustics, counting maps and semi-classical asymptotics
[article]

2010
*
arXiv
*
pre-print

The coefficients

arXiv:0912.1904v2
fatcat:2eepsdzmjrgk3jg5crrloyu3qu
*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]. ...##
###
On the Transcendence of Period Images
[article]

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 ...

##
###
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

doi:10.5860/choice.51-3296
fatcat:xm5l6ieafzewtfpuegkxsbtzhy
*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 . ...##
###
The summer meeting in ann Arbor

1955
*
Bulletin of the American Mathematical Society
*

The present paper gives

doi:10.1090/s0002-9904-1955-09979-8
fatcat:4rmgfntcavcclkvmixl6xgqg34
*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 ...
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