Leonard Lipshitz

Professor Emeritus

lipshitz@purdue.edu

Research Interest(s):

model theory, algebra, non-archimedean analysis

Recent papers

• (with Z. Robinson) Uniform properties of rigid subanalytic sets , Trans. AMS 357 (2005), no. 11, 4349-4377.
• (with Z. Robinson) Flattening and analytic continuation of affinoid morphisms. Remarks on a paper of T.S. Gardener and H. Schoutens: "Flattening and subanalytic sets in rigid analytic geometry" , [Proc. London Math. Soc. (3) 83 (2001), n.3, 681-707], Proc. London Math. Soc. (3) 91 (2005), no. 2, 443-458.
• (with R. Cluckers and Z. Robinson) Analytic Cell Decomposition and Analytic Motivic Integration , Ann. Sci. École Norm. Sup. (4) 39, (2006), 535-568.
• (with Z. Robinson) Overconvergent Real Closed Quantifier Elimination , Bulletin of the London Mathematical Society Vol. 38 Part 6, (December 2006), 897-906.
• (with R. Cluckers and Z. Robinson) Real Closed Fields with Nonstandard and Standard Analytic Structure , Journal of the London Mathematical Society Vol. 78, (2008), 198-212.
• (with R. Cluckers) Fields with Analytic Structure , Journal of the European Mathematical Society Vol 13, (2011), 1147-1223
• (with R. Cluckers) Strictly Convergent Analytic Structures , Journal of the European Mathematical Society, Vol 19 (2017) 107 - 149.

Conference Proceedings

• (J. Denef, L. Lipshitz, T. Pheidas, J. van Geel, editors) Hilbert's Tenth Problem: Relations with Arithmetic and Algebraic Geometry. A.M.S. Contemporary Mathematics 270 (2000), 367 pp.

Publications

• [1] L. Lipshitz and Z. Robinson. Flattening and analytic continuation of affinoid morphisms. Remarks on a paper of T. S. Gardener and H. Schoutens: “Flattening and subanalytic sets in rigid analytic geometry” [Proc. London Math. Soc. (3) 83 (2001), no. 3, 681-707; mr1851087]. Proc. London Math. Soc. (3), 91(2):443-458, 2005.
• [2] Leonard Lipshitz and Zachary Robinson. Uniform properties of rigid subanalytic sets. Trans. Amer. Math. Soc., 357(11):4349-4377 (electronic), 2005.
• [3] Jan Denef, Leonard Lipshitz, Thanases Pheidas, and Jan Van Geel, editors. Hilbert's tenth problem: relations with arithmetic and algebraic geometry, volume 270 of Contemporary Mathematics. American Mathematical Society, Providence, RI, 2000. Papers from the workshop held at Ghent University, Ghent, November 2-5, 1999.
• [4] Leonard Lipshitz and Zachary Robinson. Rings of separated power series and quasi-affinoid geometry. Astérisque, (264):vi+171, 2000.
• [5] Leonard Lipshitz and Zachary Robinson. Dimension theory and smooth stratification of rigid subanalytic sets. In Logic Colloquium '98 (Prague), volume 13 of Lect. Notes Log., pages 302-315. Assoc. Symbol. Logic, Urbana, IL, 2000.
• [6] Leonard Lipshitz and Zachary Robinson. Rigid subanalytic subsets of curves and surfaces. J. London Math. Soc. (2), 59(3):895-921, 1999.
• [7] L. Lipshitz and Z. Robinson. One-dimensional fibers of rigid subanalytic sets. J. Symbolic Logic, 63(1):83-88, 1998.
• [8] Leonard Lipshitz and Zachary Robinson. Rigid subanalytic subsets of the line and the plane. Amer. J. Math., 118(3):493-527, 1996.
• [9] Leonard Lipshitz and Thanases Pheidas. An analogue of Hilbert's tenth problem for p-adic entire functions. J. Symbolic Logic, 60(4):1301-1309, 1995.
• [10] L. Lipshitz. Rigid subanalytic sets. Amer. J. Math., 115(1):77-108, 1993.
• [11] Leonard Lipshitz and Lee A. Rubel. A gap theorem for differentially algebraic power series. In Number theory (New York, 1989/1990), pages 211-214. Springer, New York, 1991.
• [12] Leonard Lipshitz and Alfred J. van der Poorten. Rational functions, diagonals, automata and arithmetic. In Number theory (Banff, AB, 1988), pages 339-358. de Gruyter, Berlin, 1990.
• [13] J. Denef and L. Lipshitz. Decision problems for differential equations. J. Symbolic Logic, 54(3):941-950, 1989.
• [14] L. Lipshitz. D-finite power series. J. Algebra, 122(2):353-373, 1989.
• [15] Leonard Lipshitz and Lee A. Rubel. Corrigendum to: “A differentially algebraic replacement theorem, and analog computability” [Proc.Amer.Math.Soc.99 (1987), no.2, 367-372; MR0870803 (88e:12006)]. Proc. Amer. Math. Soc., 104(2):668, 1988.
• [16] L. Lipshitz. p-adic zeros of polynomials. J. Reine Angew. Math., 390:208-214, 1988.
• [17] L. Lipshitz. Isolated points on fibers of affinoid varieties. J. Reine Angew. Math., 384:208-220, 1988.
• [18] L. Lipshitz. The diagonal of a D-finite power series is D-finite. J. Algebra, 113(2):373-378, 1988.
• [19] J. Denef and L. Lipshitz. Algebraic power series and diagonals. J. Number Theory, 26(1):46-67, 1987.
• [20] Leonard Lipshitz and Lee A. Rubel. A differentially algebraic replacement theorem, and analog computability. Proc. Amer. Math. Soc., 99(2):367-372, 1987.
• [21] Leonard Lipshitz and Lee A. Rubel. A gap theorem for power series solutions of algebraic differential equations. Amer. J. Math., 108(5):1193-1213, 1986.
• [22] J. Denef and L. Lipshitz. Power series solutions of algebraic differential equations. Math. Ann., 267(2):213-238, 1984.
• [23] Joseph Becker, J. Denef, and L. Lipshitz. The approximation property for some 5-dimensional Henselian rings. Trans. Amer. Math. Soc., 276(1):301-309, 1983.
• [24] Leonard Lipshitz. Some remarks on the Diophantine problem for addition and divisibility. In Proceedings of the Model Theory Meeting (Univ. Brussels, Brussels/Univ. Mons, Mons, 1980), volume 33, pages 41-52, 1981.
• [25] Joseph Becker and Leonard Lipshitz. Errata to the paper: “Remarks on the elementary theories of formal and convergent power series”. Fund. Math., 112(3):241, 1981.
• [26] J. Becker, J. Denef, and L. Lipshitz. Further remarks on the elementary theory of formal power series rings. In Model theory of algebra and arithmetic (Proc. Conf., Karpacz, 1979), volume 834 of Lecture Notes in Math., pages 1-9. Springer, Berlin, 1980.
• [27] J. Denef and L. Lipshitz. Ultraproducts and approximation in local rings. II. Math. Ann., 253(1):1-28, 1980.
• [28] Joseph Becker and Leonard Lipshitz. Remarks on the elementary theories of formal and convergent power series. Fund. Math., 105(3):229-239, 1979/80.
• [29] Joseph Becker, J. Denef, L. Lipshitz, and L. van den Dries. Ultraproducts and approximations in local rings. I. Invent. Math., 51(2):189-203, 1979.
• [30] L. Lipshitz. Diophantine correct models of arithmetic. Proc. Amer. Math. Soc., 73(1):107-108, 1979.
• [31] J. Denef and L. Lipshitz. Diophantine sets over some rings of algebraic integers. J. London Math. Soc. (2), 18(3):385-391, 1978.
• [32] L. Lipshitz. Undecidable existential problems for addition and divisibility in algebraic number rings. Trans. Amer. Math. Soc., 241:121-128, 1978.
• [33] Leonard Lipshitz and Mark Nadel. The additive structure of models of arithmetic. Proc. Amer. Math. Soc., 68(3):331-336, 1978.
• [34] L. Lipshitz. The Diophantine problem for addition and divisibility. Trans. Amer. Math. Soc., 235:271-283, 1978.
• [35] Leonard Lipshitz. Undecidable existential problems for addition and divisibility in algebraic number rings. II. Proc. Amer. Math. Soc., 64(1):122-128, 1977.
• [36] L. Lipshitz. The real closure of a commutative regular f-ring. Fund. Math., 94(3):173-176, 1977.
• [37] L. Lipshitz. Integral closures of uncountable commutative regular rings. Proc. Amer. Math. Soc., 56:19-23, 1976.
• [38] Joseph Becker and Leonard Lipshitz. An application of logic to analysis. Canad. J. Math., 28(1):83-91, 1976.
• [39] L. Lipshitz. Commutative regular rings with integral closure. Trans. Amer. Math. Soc., 211:161-170, 1975.
• [40] L. Lipshitz. The undecidability of the word problems for projective geometries and modular lattices. Trans. Amer. Math. Soc., 193:171-180, 1974.
• [41] L. Lipshitz and D. Saracino. The model companion of the theory of commutative rings without nilpotent elements. Proc. Amer. Math. Soc., 38:381-387, 1973.