Some applications of Nevanlinna theory to mathematical logic: identities of exponential functions

Henson, C. Ward; Rubel, Lee A.

doi:10.1090/s0002-9947-1984-0728700-x

Cited by 32 publications

(42 citation statements)

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“…and 7 = ü then the identity t = u can be derived from "high school algebra" identities; see [18]. The latter is stronger than the Henson-Rubel conjecture in [19] that Tarski's conjecture holds on n = {/eL| if alb occurs as a subterm of t then a is a polynomial}. Let Tj £ C[z] and VJ"=1 r, • ipj)q -0; we have to show that r¡ = 0 for 1 < J < n .…”

Section: Prefacementioning

confidence: 97%

“…In the former case it is most natural to consider functions Cm -> C represented by terms in constants, variables, and operations +,-,-, exp, and in the latter case it is most natural to consider functions Nm -> N or {R+)m -> R+ represented by terms in positive (integer) constants, variables, and operations +, •, T ■ The function classes arising from both exp and { have received significant attention. The former class is called the (complex) Shanuel class (see, e.g., [8,19]). 1 The latter classes, we think, have no established names, but their subset consisting of the functions represented by one-variable terms with positive integer constants is called the Skolem set (see [7, 11,12, 15,18,29,34,36]).…”

Section: Prefacementioning

confidence: 99%

“…In [19] C. W. Henson and L. A. Rubel considered two classes of terms of LiV) in which exponentiation is permitted only for certain bases: Z(K) = {a £ L{V)\ if bc occurs as a subterm of a then b is a constant term} (the Schanuel class), and A{V) = [a e L{V)\ if bc occurs as a subterm of a then b £ V or b is a constant term} (the Levitz class). They proved in particular (Conjectures 1 and 2 and as a consequence) Tarski's conjecture for A and asked about that for the class Yl{V) = {aei(F)| if bc occurs as a subterm of a then b is a polynomial} .…”

Section: Prefacementioning

confidence: 99%

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Detecting algebraic (in)dependence of explicitly presented functions (some applications of Nevanlinna theory to mathematical logic)

Gurevič¹

1993

Trans. Amer. Math. Soc.

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Abstract. We consider algebraic relations between explicitly presented analytic functions with particular emphasis on Tarski's high school algebra problem.The part not related directly to Tarski's high school algebra problem. Let U be a connected complex-analytic manifold. Denote by ¿?(U) the minimal field containing all functions meromorphic on U and closed under exponentiation / h-ef. Let fj € &~(U),pj e J(U) -{0} for 1 < j < m, and gk 6 &(U), qk £Jt(U)-{0} for 1 < k < n (where Ji(U) is the field of functions meromorphic on 11). Let f¡ -f¡ $ ß?(U) for i ^ j and gk -g¡ £ ßf(U) (i) We describe the algebraic structure of A and 5C , where A = {t e L\ if u Î v occurs as a subterm of / then either « is a variable or u contains no variables at all} , and £? = {t e L\ if u î v occurs as a subterm of t then u 6 A} . Of these, A is a free semiring with respect to addition and multiplication but JS" is free only as a semigroup with respect to addition. A function i £ S is called +-prime in 5 if 7 / u~ + v for all u, v e S and is called multiplicatively prime in 5 if 7 = m • v => « = 1 or v = 1 for u, v € S . A function is called (+ , -J-prime in S if it is both +-prime and multiplicatively prime in S . A function in A is said to have content 1 if it is not divisible by constants in N -{1} or by / 1 (+ , ^-primes of A . The product of functions of content 1 has content 1 . Let P be the multiplicative subsemigroup of A of functions of content 1 . Then J' as a semiring is isomorphic to the semigroup semiring A(®fPf), where each Pf is a copy of P and / ranges over the ^ 1 +-primes of Sf .Received by the editors February 1, 1987 and, in revised form, April 11, 1989 1980 Mathematics Subject Classification (1985 Revision). Primary 30B40, 32D15, 03C99, 08B99; Secondary 30D35. ""Jf" refers to "meromorphic". In particular, Jf^ = {the germs meromorphic at Q = the field of quotients of the ring ßfy, and JfiU) = { meromorphic functions on U} . Note that a meromorphic function on U is a section over U of the sheaf Jf of the quotient fields of the sheaf of rings X, i.e., is a quotient of holomorphic functions in a neighborhood of every point, and is not necessarily a quotient of holomorphic functions on U. PrefaceOur involvement with this topic began in 1981 when Gregory E. Mints attracted our attention to "Tarski's high school algebra problem". Tarski's quantifier elimination for real-closed fields nowadays is used both in pure mathematics, e.g., in semialgebraic geometry and in ordinary differential equations, (for the latter see [3, §37]), and outside it, e.g., in robotics (see [32] and references there). But Alfred Tarski was interested also in the theory of the reals equipped with a transcendental operation. The most common transcendental operations are the trigonometric ones and exponentiation. The trigonometric operations present an unpleasant perspective for a logician: they permit one to define Z c R by a first-order formula and consequently they prevent decidability of the corresponding elementary theories and allow one to d...

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Section: Prefacementioning

confidence: 97%

Section: Prefacementioning

confidence: 99%

Section: Prefacementioning

confidence: 99%

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Detecting algebraic (in)dependence of explicitly presented functions (some applications of Nevanlinna theory to mathematical logic)

Gurevič¹

1993

Trans. Amer. Math. Soc.

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“…Nevanlinna theory has applications and analogies in many different fields of mathematics, such as differential equations [27,34,62,40,79], difference equations [121,128,129] number theory [98,99,115,127], Brownian motion [18] and even mathematical logic [60]. Recently, there has been increasing interest in applying Nevanlinna theory to study meromorphic solutions of complex difference equations [20,21,48,58,67,80], and in particular, to detect integrability in discrete equations [1,49,50,111].…”

Section: Nevanlinna Theorymentioning

confidence: 99%

“…where p, q ∈ S(w), or equation (60) can be transformed by a linear change in w to one of the following equations:…”

Section: Difference Equations Of Painlevé Typementioning

confidence: 99%

Meromorphic solutions of difference equations, integrability and the discrete Painlevé equations

Halburd

Korhonen

2007

J. Phys. A: Math. Theor.

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The Painlevé property is closely connected to differential equations that are integrable via related iso-monodromy problems. Many apparently integrable discrete analogues of the Painlevé equations have appeared in the literature. The existence of sufficiently many finite-order meromorphic solutions appears to be a good analogue of the Painlevé property for discrete equations, in which the independent variable is taken to be complex. A general introduction to Nevanlinna theory is presented together with an overview of recent applications to meromorphic solutions of difference equations and the difference and q-difference operators. New results are presented concerning equations of the form w(z + 1)w(z − 1) = R(z, w), where R is rational in w with meromorphic coefficients.

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Arithmetical Identities in a 2-element Model of Tarski's System

Asatryan

2002

MLQ - Math. Log. Quart.

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Some applications of Nevanlinna theory to mathematical logic: identities of exponential functions

Cited by 32 publications

References 19 publications

Detecting algebraic (in)dependence of explicitly presented functions (some applications of Nevanlinna theory to mathematical logic)

Detecting algebraic (in)dependence of explicitly presented functions (some applications of Nevanlinna theory to mathematical logic)

Meromorphic solutions of difference equations, integrability and the discrete Painlevé equations

Arithmetical Identities in a 2-element Model of Tarski's System

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