研究兴趣广泛，涉及代数数论、椭圆曲线、有限域理论、多项式理论、算术动力系统、线性递归序列、数论中的图论问题等等。研究成果丰富，至今发表了50余篇SCI论文，发表的期刊包括：Trans Amer Math Soc, Math Comp, Int Math Res Notices, J Comb Theory B, Math Zeit, Moscow Math J, Canadian J Math, Math Res Letters, Proc Amer Math Soc, Bull London Math Soc, J Algebra, J Number Theory, Acta Arith, Ramanujan J, Finite Fields Th App, J Complexity等知名期刊。
科研论文：
[59] X. Li and M. Sha, On Euler's totient function of polynomials over finite fields, Preprint, 2024.
https://arxiv.org/abs/2401.17727
[58] A. Bérczes, Y. Bugeaud, K. Győry, J. Mello, A. Ostafe and M. Sha, Explicit bounds for the solutions of superelliptic equations over number fields, Forum Mathematicum, to appear.
https://arxiv.org/abs/2310.09704
[57] A. Bérczes, Y. Bugeaud, K. Győry, J. Mello, A. Ostafe and M. Sha, Multiplicative dependence of rational values modulo approximate finitely generated groups, Mathematical Proceedings of the Cambridge Philosophical Society, to appear.
https://arxiv.org/abs/2107.05371
[56] Z. Chen, M. Sha and C. Wei, On the generalized Fibonacci sequence of polynomials over finite fields, Preprint, 2023.
http://arxiv.org/abs/2303.17525
[55] X. Li and M. Sha, On the Lucas and Lehmer sequences in transcendental Dedekind domains, Publicationes Mathematicae Debrecen, to appear.
https://arxiv.org/abs/2006.09880
[54] A. Dubickas, M. Sha and I. E. Shparlinski, Euclidean minima of algebraic number fields, Archiv der Mathematik, 122 (2024), 405-414.
http://arxiv.org/abs/2307.10880
[53] S. Hu, M. Kim and M. Sha, On the congruences of Eisenstein series with polynomial indexes, Ramanujan Journal, 62 (2023), 413-430.
https://arxiv.org/abs/1805.09225
[52] B. Mans, M. Sha, I. E. Shparlinski and D. Sutantyo, Functional graphs of families of quadratic polynomials, Mathematics of Computation, 92(2023), 2307-2331.
http://arxiv.org/abs/2208.01885
[51] S. V. Konyagin, M. Sha, I. E. Shparlinski and C. L. Stewart,
On the distribution of multiplicatively dependent vectors, Mathematical Research Letters, 30 (2023), 509-540.
https://arxiv.org/abs/1903.09796
[50] A. Dubickas and M. Sha,
Counting decomposable polynomials with integer coefficients, Monatshefte fur Mathematik, 200 (2023), 229-253.
https://arxiv.org/abs/1803.08755
[49] S. Bae, S. Hu and M. Sha,
On the Carmichael rings, Carmichael ideals and Carmichael polynomials, Colloquium Mathematicum, 171 (2023), 1-17.
https://arxiv.org/abs/1809.05432
[48] F. Barroero, L. Capuano, L. Merai, A. Ostafe and M. Sha,
Multiplicative and linear dependence in finite fields and on elliptic curves modulo primes, International Mathematics Research Notices, 2022 (2022), 16094-16137.
https://arxiv.org/abs/2008.00389
[47] J. Mello and M. Sha,
On the properties of Northcott and Narkiewicz for elliptic curves, International Journal of Number Theory, 18 (2022), 2129-2144.
https://arxiv.org/abs/1911.08752
[46] M. Sha,
Zsigmondy's theorem and primitive divisors of the Lucas and Lehmer sequences in polynomial rings, Journal of Algebra, 586 (2021), 830-843.
https://arxiv.org/abs/2005.01940
[45] M. Sha and I.E. Shparlinski,
Mobius randomness law for Frobenius traces of ordinary curves, Canadian Mathematical Bulletin, 64 (2021), 192-203.
https://arxiv.org/abs/1909.00969
[44] F. Barroero and M. Sha,
Torsion points with multiplicatively dependent coordinates on elliptic curves, Bulletin of the London Mathematical Society, 52 (2020), 807-815.
https://arxiv.org/abs/1904.02474
[43] X. Li and M. Sha,
A proof of Sondow's conjecture on the Smarandache function, The American Mathematical Monthly, 127 (2020), 939-943.
https://arxiv.org/abs/1907.00370
[42] X. Li and M. Sha,
Polynomial analogue of the Smarandache function, Journal of Number Theory, 217 (2020), 320-339.
https://arxiv.org/abs/1906.00510
[41] B. Mans, M. Sha, J. Smith and D. Sutantyo,
On the equational graphs over finite fields, Finite Fields and Their Applications, 64 (2020), Article 101667.
https://arxiv.org/abs/1906.12054
[40] X. Li and M. Sha,
Congruence preserving functions in the residue class rings of polynomials over finite fields, Finite Fields and Their Applications, 61 (2020), Article 101604.
https://arxiv.org/abs/1807.02379
[39] S. Hu, M. Kim, P. Moree and M. Sha,
Irregular primes with respect to Genocchi numbers and Artin's primitive root conjecture, Journal of Number Theory, 205 (2019), 59-80.
https://arxiv.org/abs/1809.08431
[38] P. Moree and M. Sha,
Primes in arithmetic progressions and nonprimitive roots, Bulletin of the Australian Mathematical Society, 100 (2019), 388-394.
https://arxiv.org/abs/1901.02650
[37] X. Li and M. Sha,
Polynomial functions in the residue class rings of Dedekind domains, International Journal of Number Theory, 15 (2019), 1473-1486.
https://arxiv.org/abs/1704.04965
[36] B. Mans, M. Sha, I.E. Shparlinski and D. Sutantyo,
On functional graphs of quadratic polynomials, Experimental Mathematics, 28 (2019), 292-300.
https://arxiv.org/abs/1706.04734
[35] A. Ostafe, M. Sha, I.E. Shparlinski and U. Zannier,
On multiplicative dependence of values of rational functions and a generalisation of the Northcott theorem, Michigan Mathematical Journal, 68 (2019), 385-407.
https://arxiv.org/abs/1706.05874
[34] S. Hu and M. Sha,
On the additive and multiplicative structures of the exceptional units in finite commutative rings, Publicationes Mathematicae Debrecen, 94 (2019), 369-380.
https://arxiv.org/abs/1612.04539
[33] M. Sha,
Effective results on the Skolem Problem for linear recurrence sequences, Journal of Number Theory, 197 (2019), 228-249.
https://arxiv.org/abs/1505.07147
[32] A. Dubickas and M. Sha,
Multiplicative dependence of the translations of algebraic numbers, Revista Matematica Iberoamericana, 34 (2018), 1789-1808.
https://arxiv.org/abs/1608.05458
[31] A. Dubickas and M. Sha,
The distance to square-free polynomials, Acta Arithmetica, 186 (2018), 243-256.
https://arxiv.org/abs/1801.01240
[30] D. Gomez-Perez, M. Sha and A. Tirkel,
On the linear complexity for multidimensional sequences, Journal of Complexity, 49 (2018), 46-55.
https://arxiv.org/abs/1803.03912
[29] R. de la Breteche, M. Sha, I.E. Shparlinski and J.F. Voloch,
The Sato-Tate distribution in thin parametric families of elliptic curves, Mathematische Zeitschrift, 290 (2018), 831-855.
https://arxiv.org/abs/1509.03009
[28] F. Pappalardi, M. Sha, I.E. Shparlinski and C. Stewart,
On multiplicatively dependent vectors of algebraic numbers, Transactions of the American Mathematical Society, 370 (2018), 6221-6244.
https://arxiv.org/abs/1606.02874
[27] A. Ostafe, M. Sha, I.E. Shparlinski and U. Zannier,
On abelian multiplicatively dependent points on a curve in a torus, Quarterly Journal of Mathematics, 69 (2018), 391-401.
https://arxiv.org/abs/1704.04694
[26] M. Sha and I.E. Shparlinski,
Effective results on linear dependence for elliptic curves, Pacific Journal of Mathematics, 295 (2018), 123-144.
https://arxiv.org/abs/1410.1596
[25] A. Dubickas and M. Sha,
On the number of integer polynomials with multiplicatively dependent roots, Acta Mathematica Hungarica, 154 (2018), 402-428.
https://arxiv.org/abs/1707.04965
[24] D. Gomez-Perez, A. Ostafe and M. Sha,
The arithmetics of consecutive polynomial sequences over finite fields, Finite Fields and Their Applications, 50 (2018), 35-65.
https://arxiv.org/abs/1509.01936
[23] A. Dubickas, M. Sha and I.E. Shparlinski,
On distances in lattices from algebraic number fields, Moscow Mathematical Journal, 17 (2017), 239-268.
https://arxiv.org/abs/1703.02163
[22] F. Luca, M. Sha and I.E. Shparlinski,
On two functions arising in the study of Carmichael quotients, Colloquium Mathematicum, 149 (2017) , 179-192.
https://arxiv.org/abs/1705.00388
[21] X. Li and M. Sha,
Gauss factorials of polynomials over finite fields, International Journal of Number Theory, 8 (2017), 2039-2054.
https://arxiv.org/abs/1704.04972
[20] M. Sha and I.E. Shparlinski,
The Sato-Tate distribution in families of elliptic curves with a rational parameter of bounded height, Indagationes Mathematicae, 28 (2017), 306-320.
https://arxiv.org/abs/1512.07301
[19] A. Ostafe and M. Sha,
Counting dynamical systems over finite fields, Contemporary Mathematics, 669 (2016), 187-203.
https://arxiv.org/abs/1505.03618
[18] S.V. Konyagin, F. Luca, B. Mans, L. Mathieson, M. Sha and I.E. Shparlinski,
Functional graphs of polynomials over finite fields, Journal of Combinatorial Theory, Series B, 116 (2016), 87-122.
https://arxiv.org/abs/1307.2718
[17] A. Dubickas and M. Sha,
Positive density of integer polynomials with some prescribed properties, Journal of Number Theory, 159 (2016), 27-44.
https://arxiv.org/abs/1504.05144
[16] M. Sha and I.E. Shparlinski,
Lang-Trotter and Sato-Tate distributions in single and double parametric families of elliptic curves, Acta Arithmetica, 170 (2015), 299-325.
https://arxiv.org/abs/1404.0182
[15] A. Dubickas, M. Sha and I.E. Shparlinski,
Explicit form of Cassels' p-adic embedding theorem for number fields, Canadian Journal of Mathematics, 67 (2015), 1046-1064.
https://arxiv.org/abs/1401.6819
[14] A. Ostafe and M. Sha,
On the quantitative dynamical Mordell-Lang conjecture, Journal of Number Theory, 156 (2015), 161-182. Corrigendum: Journal of Number Theory, 164 (2016), 433-437.
https://arxiv.org/abs/1501.02543
[13] A. Dubickas and M. Sha,
Counting and testing dominant polynomials, Experimental Mathematics, 24 (2015), 312-325.
https://arxiv.org/abs/1407.2789
[12] M. Sha,
On the lattices from elliptic curves over finite fields, Finite Fields and Their Applications, 31 (2015), 84-107.
https://arxiv.org/abs/1406.3086
[11] M. Sha,
The arithmetic of Carmichael Quotients, Periodica Mathematica Hungarica, 71 (2015), 11-23. Corrigendum: Periodica Mathematica Hungarica, https://doi.org/10.1007/s10998-017-0227-7.
https://arxiv.org/abs/1108.2579
[10] A. Dubickas and M. Sha,
Counting degenerate polynomials of fixed degree and bounded height, Monatshefte fur Mathematik, 177 (2015), 517-537.
https://arxiv.org/abs/1402.5430
[9] M. Sha,
On the non-idealness of cyclotomic families of pairing-friendly elliptic curves, Journal of Mathematical Cryptology, 8 (2014), 417-440.
https://arxiv.org/abs/1304.7169
[8] M. Sha,
Heuristics of the Cocks-Pinch method, Advances in Mathematics of Communications, 8 (2014), 103-118.
https://arxiv.org/abs/1211.0971
[7] M. Sha,
Bounding the j-invariant of integral points on certain modular curves, International Journal of Number Theory, 10 (2014), 1545-1551.
https://arxiv.org/abs/1210.3224
[6] M. Sha,
Bounding the j-invariant of integral points on modular curves, International Mathematics Research Notices, 2014 (2014), 4492-4520.
https://arxiv.org/abs/1208.1337
[5] A. Bajolet and M. Sha,
Bounding the j-invariant of integral points on X_{ns}^{+}(p), Proceedings of the American Mathematical Society, 142 (2014), 3395-3410.
https://arxiv.org/abs/1203.1187
[4] M. Sha,
Digraphs from endomorphisms of finite cyclic groups, Journal of Combinatorial Mathematics and Combinatorial Computing, 83 (2012), 105-120.
https://arxiv.org/abs/1007.1712
[3] M. Sha and L. Yin,
Galois groups and genera of a kind of quasi-cyclotomic function fields, Journal of Number Theory, 132 (2012), 2574-2581.
https://arxiv.org/abs/1007.1729
[2] M. Sha and S. Hu,
Monomial dynamical systems of dimension one over finite fields, Acta Arithmetica, 148 (2011), 309-331.
https://arxiv.org/abs/0910.5550
[1] M. Sha,
On the cycle structure of repeated exponentiation modulo a prime power, Fibonacci Quarterly, 49 (2011), 340-347.
https://arxiv.org/abs/1101.3482 