CURRICULUM VITAE

MARIAN VAJAITU

Born:    September 27, 1962, Colt, Romania.
Studies: I  graduated from Department of Mathematics of  Mathematics of
the University of Bucharest in 1987. The title of Diploma
Dissertation was "Riemann's zeta  function and applications in
Number Theory".

The main lectures during this period are: two year a basic course in algebra, an year a basic course in real analysis, complex analysis (a semester), measure theory (a semester), two years a basic course in differential equations, an year of axiomatic and euclidean geometry, an year of differential geometry, an year of functional analysis, a supplementary year of algebra (number theory and groups theory), a course in Algebraic Geometry and one in Algebraic Topology.

I studied some classical works in number theory, such as:

   - analytic number theory  (from the books by Ingham, Eduards,
Davenport, Titchmarsh, Halberstan, Baker, Bump).
- algebraic number theory (from the books by Borevich and Shafarevich,
H. Hasse and S. Lang).
- valuations and the theory of algebraic functions (from the books by
Schilling, Chevalley and Hasse).
- class field theory (from the books by Hasse, Chevalley and Neurkirch).


I participated also at the seminar of Number Theory lead by Professor N. Popescu devoted to algebraic number theory, analytic number theory, class field theory and theory of algebraic functions.

Employment:  1987-1990 -  Teacher at the High School at Petrosani, Romania
(forced by the laws of former regime),
since 1990 - Researcher at Institute of  Mathematics,
Bucharest, Romania.

Fields of interest: Number theory with special interest in analytic number
theory, algebraic numbers and functions,
transcendental numbers, local class field theory,
diofantine equations.


Scientific activity: Working in the field of Number Theory, I obtained some results. Part of them are collected in some papers, from which I mention:

• A result dealing with some arithmetical conjectures: The conjecture of Graham " for every chain of integers 0 < a_1 < ... < a_n" is solved for large numbers "n" by A. Zaharescu. Assuming the Riemann Hypothesis we proved that Graham's statement is true for any n\geq 10^70, in Journal of Number Theory, vol. 31, No. 1, january 1989.
• A problem of Selfridge which asks for the pairs (a, b) for which 2^a - 2^b | n^a - n^b for all n, INCREST preprint series in mathematics, No. 32/1989, by finding the 14 such pairs with a > b. I have to say that we have seen that this problem was already solved by two (Chinese) mathematicians a few years ago. On the other hand in this preprint it is proved also that, more generally, under some simple assumptions, given a_1,...,a_k\in\ZZ^*, the set of (k + 1)-uples (\beta,(\alpha_1,...,\alpha_k )) such that \Sum_{i=1}^{i=k}(\aplha_i)\times(\beta)^{a_i} divides \Sum_{i=1}^{i=k}(\aplha_i)\times n^{a_i} for all n, is finite.
• Doctoral thesis in 1994 "Estimations of the ideal generated by the values of a polynomial over a Dedekind ring".
• A finiteness theorem for a class of exponential congruences (to appear in Proc. A.M.S.).
• The ideal generated by the values of a polynomial over a Dedekind ring, Rev. Roumaine Math. Pures Appl.,42 (1997), 155-161.
• An inequality involving the degree of an algebraic set, Rev. Roumaine Math. Pures Appl., 43(1998), 451-455.
• Uniform distributions in local fields (to appear).

AWARDS: 2001 PRIZE GH. LAZAR'' OF THE ROMANIAN ACADEMY FOR MATHEMATICS.

LIST OF PAPERS

1. Graham's Conjecture under Riemann Hypothesis, J. Number Theory, vol.31, No.1, 1989. (With C. Cobeli and A. Zaharescu.) MR0978101 (90c:11064).
2. Doctoral thesis, Estimations of the ideal generated by the values of a polynomial over a Dedekind ring, 1994.
3. The ideal generated by the values of a polynomial over a Dedekind ring, Rev. Roumaine math. pures appl., 42(1997), 1--2, 155--161. MR1650032 (99m:13037).
4. An inequality involving the degree of an algebraic set, Rev. Roumaine math. pures appl., 43(1998), 3--4, 451--455. MR1845541 (2002e:13039).
5. A finiteness theorem for a class of exponential congruences, Proc. of the American Math. Soc., 127(1999), no.8, 2225--2232. (With A. Zaharescu.) MR1486757 (99j:11003).
6. The sequence $n!\pmod p$, Journal of the Ramanujan Math. Soc., 15(2000), no.2, 71--90. (With C. Cobeli and A. Zaharescu.) MR1754715 (2001g:11153).
7. Average estimates for the number of tuples of inverses $\pmod p$ in short intervals, Bull. Math. Soc. Sc. Math. Roumanie, Thome 43(91), no.2, 2000, 155--167. (With C. Cobeli and A. Zaharescu.) MR1881326 (2002k:11168).
8. Uniform distributions in $p-$adic fields. An Erdos--T\'uran inequality, Buletin Stiintific--Univ. Pitesti, Seria Matematica si Informatica, Nr.5(2000), 1--7. (With A. Zaharescu.)
9. The analytic continuation problem of $p-$adic $L-$functions, Math. Reports, vol.2(52), no.3(2000), 379--389. (With A. Zaharescu.) MR1889620 (2003b:11130).
10. Exponential sums and their role in number theory (I), Rev. Roumaine 45, nr.6(2000), 1035--1049. (With A. Zaharescu.) MR1886536 (2002m:11076).
11. An Erdos--T\'uran inequality in $p-$adic fields, Math. Reports, vol.2(52), no.2(2000), 243--252. (With A. Zaharescu.) MR1890979 (2003a:11093).
12. Integer points unusually close to elliptic curves, Math. Portugaliae vol.58, Fasc.2(2001), 211--218. (With A. Zaharescu.) MR1836263 (2002e:11047).
13. Equidistribution of rational functions of primes $\pmod q$, Journal of the Ramanujan Math. Soc. vol.16, no.1(2001), 63--73. (With C. Cobeli and A. Zaharescu.) MR1824884 (2002b:11102).
14. Integer points near hyperelliptic curves, C.R.Math.Rep. Acad.Sci. Canada, vol.23(2001), 84--90. (With A. Zaharescu.) MR1852803 (2002e:11085).
15. $L-$functions associated to Galois orbits in $\CC_p$, Math. Reports, vol.3(53), nr.1(2001), 83--89. (With A. Zaharescu.) MR1887188 (2003d:11175).
16. Groups of isometries on ultrametric spaces, Bull. Math. Soc. Sc. Math. Roumanie, Thome 44(92), no.2(2001), 183--191. (With A. Zaharescu.) MR2015104 (2004i:20006).
17. The change of space problem for metric locally constant functions, Buletin Stiintific-Univ. Pitesti, Seria Matematica si Informatica, Nr.7(2001), 179--183. (With A. Zaharescu.)
18. Galois groups with metric constraints, Bull. Math. Sci. Roumaine, vol.44(92), no.3(2001), 211--219. (With A. Zaharescu.) MR2013339 (2004g:11105).
19. Uniform distribution of polynomial sequences in $p-$adic fields, Buletin Stiintific--Univ. Pitesti, Seria Matematica si Informatica, no.7(2001). (With A. Zaharescu.)
20. Exponential sums and their role in number theory (II), Rev. Roumaine 47(2002), 135--148.(With A. Zaharescu.) MR1978194 (2004e:11084).
21. Integer points close to algebraic curves, Journal of the L.M.S., vol.65(2002), no.1, 10--26. (With F.P. Boca and A. Zaharescu.) MR1875132 (2003b:11067).
22. On the set $ax+bg^x \pmod p$, Math. Portugaliae, vol.59, Fasc.2(2002), 195--204. (With C. Cobeli and A. Zaharescu.) MR1907414 (2003d:11113).
23. Differences between powers of a primitive root, Internat. Journal Math. Sci., vol.29, no.6(2002), 325--331. (With A. Zaharescu.) MR1897859 (2003e:11002).
24. Chains of metric invariants over $p-$adic fields, Acta Arithmetica, 103, 1(2002), 27--40. (With A. Popescu, N. Popescu and A. Zaharescu.) MR1904892 (2003g:11138).
25. Generalization of a theorem of Steinhauss, Colloq. Math., vol.92(2002), no. 1--2, 257--266. (With C. Cobeli, G. Groza and A. Zaharescu.) MR1899442 (2003h:11082).
26. Distribution of values of rational maps on the ${\bf F}_p-$points on an affine curve, Monatshefte fur Mathematik, 136(2002), 81--86. (With A. Zaharescu.) MR1908082 (2003f:11089).
27. Character sums and pair correlations, Demonstratio Math., vol.35, no.2(2002), 225--232. (With A. Zaharescu.) MR1907295 (2003d:11123).
28. A class of algebraic--exponential congruences modulo $p$, Acta Math. Univ. Comenianae, 71(2002), no.1, 113--117. (With C. Cobeli and A. Zaharescu.) MR1943018 (2003m:11208).
29. A class of irreducible polynomials, Journal of the Ramanujan Math. Soc. vol.17(2002), no.3, 161--172. (With M. Cavachi and A. Zaharescu.) MR1925187 (2003g:12002).
30. Polynomial--exponential equations modulo $p$, Buletin Stiintific-Univ. Pitesti, Seria Matematica si Informatica, Nr.8(2002), 183--188. (With C. Cobeli and A. Zaharescu.)
31. Metric locally constant functions, Acta et Commentationes Universitatis Tartuensis de Mathematica, vol.6(2002), 29--36. (With A. Zaharescu.) MR1962874 (2004d:11117).
32. Distinct gaps between fractional parts of sequences, Proc. A.M.S., vol.130, no.12(2002), 3447--3452. (With A. Zaharescu.) MR1918819 (2003d:11112).
33. Distribution of gaps between the inverses $\pmod q$, Proc. Edinburgh Math. Soc., 46(2003), 1--19. (With C. Cobeli and A. Zaharescu.) MR1961820 (2004a:11105).
34. Sequences of values of power series over $p-$adic fields, Revue Roumaine, 48(2003), 2, 211--216. (With A. Zaharescu.) MR1999021 (2004i:11089).
35. Nonarchimedean valuation on $\RR$ and $\CC$, Math. Reports, 5(55), 3(2003), 267--273. (With A. Zaharescu.)
36. On the unicity of immediate maximal extensions of valued fields, Math. Journal of Ibaraki Univ., vol.35(2003), 29--33. (With A. Zaharescu.) MR2040538.
37. An irreducibility criterion for polynomials in several variables, Acta Math. et Informatica Univ. Ostraviensis, to appear. (With M. Cavachi and A. Zaharescu.)
38. On the existence of trace for elements of $\CC_p$, Algebras and Representation Theory, to appear. (With N. Popescu and A. Zaharescu.)
39. Some asymptotic formulas involving primes in arithmetic progression, Commentarii Math. Univ. St. Pauli, Japan, Vol. 53, No. 1(2004), 23--35. (With C. Cobeli, L. Panaitopol and A. Zaharescu.)
40. Regularization on ordered sheaves, Revue Roumaine math. pures appl., {\bf 43}(2004), 3, 303--309. (With A. Zaharescu.)
41. On Krasner analytic functions and the $p-$adic Mellin transform, Math. Journal of Ibaraki Univ. submisa. (With A. Zaharescu.)
42. Transformation formulas for $L-$functions associated with Galois orbits in $\CC_p$, Revue Roumaine, to appear. (With A. Zaharescu.)
43. Primitive arcs on elliptic curves, Revue Roumaine, to appear.