I prove a version of Schanuel's conjecture for Weierstrass equations in differential fields, answering a question of Zilber, and show that the linear independence condition in the statement cannot be relaxed

Jonathan Kirby

Kirby

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Kirby, Jonathan

University of East Anglia digital repository

Previous Up Next ArticleMR2140050 (2006c:12010a) 12H05 (03C60)Kirby, Jonathan (4-OX)A Schanuel condition for Weierstrass equations. (English summary)J. Symbolic Logic 70 (2005), no. 2, 631–638. CitationsFrom References: 0From Reviews: 0MR2158459 (2006c:12010b) 12H05 (03C60)Kirby, Jonathan (4-OX)Corrigendum to: “A Schanuel condition for Weierstrass equations” [J. Symbolic Logic 70(2005), no. 2, 631–638; MR2140050 (2006c:12010a)].J. Symbolic Logic 70 (2005), no. 3, 1023.Consider an ordinary differential field F of characteristic zero with differentiation δ andC as the field of constants. For n solutions in F of the equation δyi = yiδxi we havetr.deg .CC(x1, y1, . . . , xn, yn) ≥ n+1 if the elements δxi are linearly independent over C [seeJ. Ax, Ann. of Math. (2) 93 (1971), 252–268; MR0277482 (43 #3215)]. The paper under reviewstudies such a problem for another differential equation, namely, for the Weierstrass one: (δy)2 =f(y)(δx)2, where f is a cubic with constant coefficients and without multiple roots.The author first gives a similar result with one equation and many solutions (Proposition 1.1), andthen does this having several different cubics fi (Proposition 1.2) with a certain extra condition forrelations between fi. Finally, these results are generalized for a partial differential field F with thebasic set of differentiations ∆= {δ1, . . . , δs}. The statement is similar: tr.deg .CC({xik, yik})≥∑i ni+ r, where r is the rank of the Jacobi matrix (δlxik) and (xik, yik) are ni solutions for thecorresponding Weierstrass equations with different cubics fi.In the corrigendum the author provides corrections for the two statements of the paper underreview: the conditions for the relations between cubics fi in Propositions 1.2 and 3.2 should begiven in a stronger way.Reviewed by Alexey I. OvchinnikovReferences for 2006c:12010a1. James Ax, On Schanuel’s conjectures, Annals of Mathematics, vol. 93 (1971), pp. 252–268.MR0277482 (43 #3215)2. Cristiana Bertolin, P’eriodes de 1-motifs et transcendance, Journal of Number Theory, vol. 97(2002), pp. 204–221. MR1942957 (2003i:11104)3. W. Dale Brownawell and K. K. Kubota, The algebraic independence of Weierstrass functionsand some related numbers, Acta Arithmetica, vol. XXXIII (1977), pp. 111–149. MR0444582(56 #2932)4. Ehud Hrushovski, A new strongly minimal set, Annals of Pure and Applied Logic, vol. 63(1993), no. 2, pp. 147–166. MR1226304 (94d:03064)5. E. R. Kolchin, Galois theory of differential fields, American Journal of Mathematics, vol. 75(1953), pp. 753–824. MR0058591 (15,394a)6. David Marker, Model theory of differential fields, Model Theory of Fields (David Marker,Margit Messmer, and Anand Pillay, editors), Lecture Notes in Logic, vol. 5, Springer, 1996.7. A. Seidenberg, Abstract differential algebra and the analytic case, Proceedings of the AmericanMathematical Society, vol. 9 (1958), no. 1, pp. 159–164. MR0093655 (20 #178)8. , Abstract differential algebra and the analytic case II, Proceedings of the American Mathe-matical Society, vol. 23 (1969), no. 3, pp. 689–691. MR0248122 (40 #1376)9. Joseph Silverman, The Arithmetic of Elliptic Curves, Graduate Texts in Mathematics, vol. 106,Springer, 1986. MR0817210 (87g:11070)10. Boris Zilber, Complex geometry and pseudo-analytic structures, Oberwolfach tutorial, avail-able on http://www.maths.ox.ac.uk/∼zilber, 2004.11. , Analytic and pseudo-analytic structures, Logic colloquium 2000 (R. Cori et al., editors),Lecture Notes in Logic, vol. 19, AK Peters, 2005, pp. 392– 408. MR2143889 (2006d:03058)Note: This list, extracted from the PDF form of the original paper, may contain data conversionerrors, almost all limited to the mathematical expressions.References for 2006c:12010b1. Jonathan Kirby, A Schanuel condition for Weierstrass equations, this Journal, vol. 70 (2005),no. 3, pp. 631–638. MR2140050 (2006c:12010a)Note: This list, extracted from the PDF form of the original paper, may contain data conversionerrors, almost all limited to the mathematical expressions.c© Copyright American Mathematical Society 2006, 2011

A Schanuel condition for Weierstrass equations

Kirby, J

Corrigendum to “A Schanuel Condition for Weierstrass Equations”

I prove a version of Schanuel's conjecture for Weierstrass equations in differential fields, answering a question of Zilber, and show that the linear independence condition in the statement cannot be relaxed. © 2005. Association for Symbolic Logic

Corrigendum to “A Schanuel Condition for Weierstrass Equations”

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