Preprints, Papers, Notes

These notes are in reverse chronological order of when they were written, not in the order in which they were published. Some of them are still unfinished and have never been submitted anywhere. For a list of published works click here

• sh-107: Weighted Algebraic Codes and Their Quantum Extensions (Master's Thesis)

• sh-106: Quantum Error Correction and the Future of Operating Systems

• sh-105: Graded Transformers for Safety-Critical Autonomous Driving  (unavailable for public display)

• sh-104: Graded Quantum Codes

• sh-103:  Quantum-Resistant Algebraic Codes for Blockchain Cryptography

• sh-102: Graded Quantum Codes: From Weighted Algebraic Geometry to Homological Chain Complexes

• sh-101: Weighted Heights on Toric Varieties: A Combinatorial and Arithmetic Framework,
  Q. Gashi, T. Shaska

• sh-100: Diagonalizable weighted hypersurfaces  

• sh-99: A mathematical framework to data fabrics

• sh-98: Hitchin spinors on genus-two curves with symmetries,
  A. Malmendier, A. Clingher, T. Shaska

• sh-97: Finsler Metric Clustering in Weighted Projective Spaces,
  Acta Mathematica Hungarica 

• sh-96: Quantum Gröbner: Taming Weighted Varieties

• sh-95: Graded Transformers: A Symbolic-Geometric Approach to Structured Learning
  IEEE Transactions on Neural Networks and Learning Systems

• sh-94: Arithmetic Sparsity and Cohomological Obstructions in Weighted Projective Spaces

• sh-93: Isogenies, Kummer surfaces, and theta functions,
  NATO Sci. Peace Secur. Ser. D Inf. Commun. Secur.  Vol. 66

• sh-92: Computing Weierstrass form of superelliptic curves

• sh-91: Rational Points and Zeta Functions of Humbert Surfaces with Square Determinant over F_q,
  NATO Sci. Peace Secur. Ser. D Inf. Commun. Secur.  Vol. 66

• sh-90: Weighted Heights and GIT Heights,  
  Journal of Algebraic Combinatorics  (submitted) 

• sh-89:  Graded Neural Networks
  International Journal of Data Science in the Mathematical Sciences  (submitted) 

• sh-88: Optimization of Vector Functions Using the Max Norm

• sh-87: Rational points of weighted hypersurfaces over finite fields, S. Salami, T. Shaska

• sh-86: Neuro-Symbolic Learning for Irreducible Sextics: Unveiling Probabilistic Trends in Polynomials,
  IEEE Transactions on Artificial Intelligence

• sh-85: Galois Groups of Quintics: A Neurosymbolic Approach to Polynomial Classification,
  International Journal of Data Science in the Mathematical Sciences 

• 2024-06: A Neurosymbolic Framework for Geometric Reduction of Binary Forms,
  Contemporary Math.   2025

• 2024-05: Polynomials, Galois groups,  and Database-Driven Arithmetic,
  Contemporary Math.   2025

• 2024-04: Rational Functions on the Projective Line from a Computational Viewpoint,
  Journal of Symbolic Computation

• 2024-03: Machine learning for moduli space of genus two curves and an application to isogeny based cryptography, 
  Journal of Algebr Comb 61, 23 (2025).

• 2024-02: Artificial neural networks on graded vector spaces   
  Contemporary Math,  2025

• 2024-01: Equations for generalized superelliptic Riemann surfaces, R. Hidalgo, S. Quispe, T. Shaska

• 2023-01:   Vojta's conjecture on weighted projective varieties, S. Salami, T. Shaska;
  European Journal of Mathematics,  11, 12 (2025).  

• 2022-1: Local and global heights on weighted projective varieties, S. Salami, T. Shaska;
  Houston J. Math. Vol. 49, #3, (2023), pg. 603-636

• 2021-2: Arithmetic inflection of superelliptic curves,  E Cotterill, I Darago, C. G López, C Han, T Shaska, 
  Michigan J. Math. 

• 2021-1: Geometry of Prym varieties for certain bielliptic curves of genus three and five,
  Pure Appl. Math. Q.  17  1739--1784  (2021)  https://doi.org/10.4310/PAMQ.2021.v17.n5.a5 

• 2020-1: Reduction of superelliptic Riemann surfaces,
  Contemporary Math. 776  227--247  (2022)    https://doi.org/10.1090/conm/776/15614   

• 2020-i: Integrable systems: a celebration of Emma Previato's 65th birthday,  R. Donagi and T. Shaska, 458,1--12  (2020) 

• 2020-ii: Algebraic geometry: a celebration of Emma Previato's 65th birthday, R. Donagi and T. Shaska,  1--12  (2020)

• 2019-5: The addition on Jacobian varieties from a geometric viewpoint, Y. Kopeliovich, T. Shaska     https://arxiv.org/abs/1907.11070

• 2019-4: From hyperelliptic to superelliptic curves,  A. Malmendier and T. Shaska;
  Albanian J. Math.  13  107--200  (2019)

• 2019-3:  Superelliptic curves with many automorphisms and CM Jacobians,  A. Obus and T. Shaska
  Math. Comp.  90  2951--2975  (2021) https://doi.org/10.1090/mcom/3639 

• 2019-2: On isogenies among certain abelian surfaces, A. Clingher, A. Malmendier, T. Shaska, 
  Mich. Math. J.  71  227--269  (2022) https://doi.org/10.1307/mmj/20195790 

• 2019-1:  Weighted greatest common divisors and weighted heights,
  J. Number Theory  213  319--346  (2020)  https://doi.org/10.1016/j.jnt.2019.12.012  

• 2018-6: On automorphisms of algebraic curves,
  Contemporary Math. 724  175--212  (2019) https://doi.org/10.1090/conm/724/14590 

• 2018-5: Kay Magaard (1962--2018),  Gerhard Hiss and Tony Shaska
  Albanian J. Math.  12  33--35  (2018) https://albanian-j-math.com/archives/2018-05.pdf

• 2018-4: Computing heights on weighted projective spaces,  J. Mandili, T. Shaska
  Contemporary Math.  724  149--160  (2019) https://doi.org/10.1090/conm/724/14588   

• 2018-3:  Six line configurations and string dualities, A.Clingher, A.Malmendier, T. Shaska, 
  Comm. Math. Phys.  371  159--196  (2019) https://doi.org/10.1007/s00220-019-03372-0 

• 2018-2: On the discriminant of certain quadrinomials, Sh. Otake, T. Shaska, 
  Contemporary Math. 724  55--72  (2019) https://doi.org/10.1090/conm/724/14585   

• 2018-1: Curves, Jacobians, and cryptography, G. Frey, T. Shaska
  Contemporary Math. 724  279--344  (2019) https://doi.org/10.1090/conm/724/14596   

• 2017-4: Coing Theory, Alfred J. Menezes, Paul C. van Oorschot, David Joyner, Tony Shaska, Douglas R. Shier, Wayne Goddard,
  Chapter to Handbook of Discrete and Combinatorial Mathematics

• 2017-3: Some remarks on the non-real roots of polynomials, Sh. Otake, T. Shaska,
  Cubo  20  67--93  (2018)

• 2017-2: Isogenous components of Jacobian surfaces,  L. Beshaj, A. Elezi, T. Shaska,
  European Journal of Mathematics. Vol. 6  1276--1302  (2020)

• 2017-1:  Reduction of binary forms via the hyperbolic centroid, A. Elezi, T. Shaska, 
  Lobachevskii J. Math.  42  84--95  (2021) https://doi.org/10.1134/s199508022101011x 

• 2016-6: On generalized superelliptic Riemann surfaces, R. Hidalgo, S. Quispe, T. Shaska
  Transformation Groups (2025)       https://arxiv.org/abs/1609.09576

• 2016-5: Rational points in the moduli space of genus two,
  Contemp. Math., 703, 83--115  (2018)  https://doi.org/10.1090/conm/703/14132 

• 2016-4: The Satake sextic in F-theory,  A. Malmendier, T. Shaska, 
  J. Geom. Phys.  120  290--305  (2017)   https://doi.org/10.1016/j.geomphys.2017.06.010 

• 2016-3: A universal genus-two curve from Siegel modular forms,  A. Malmendier, T. Shaska; 
  SIGMA Symmetry Integrability Geom. Methods Appl.  13  Paper No. 089, 17  (2017) 

• 2016-2: Self-inversive polynomials, curves, and codes, D. Joyner, T. Shaska,
  Contemp. Math., 703, AMS, 2018, 189–208. https://doi.org/10.1090/conm/703/14138  

• 2016-1:  On the field of moduli of superelliptic curves, R. Hidalgo, T. Shaska,
  Contemp. Math., 703, AMS, 2018, 47–62. https://doi.org/10.1090/conm/703/14130   

• 2015-4: Theta functions of superelliptic curves,
  NATO Sci. Peace Secur. Ser. D Inf. Commun. Secur., 41, IOS Press, Amsterdam, 2015, 47–69.

• 2015-3: Weight distributions, zeta functions and Riemann hypothesis for linear and algebraic geometry codes, Elezi/Shaska
  Albanian J. Math.  41  328--359  (2015)

• 2015-2: Weierstrass points of superelliptic curves, C. Shor, T. Shaska,
  NATO Sci. Peace Secur. Ser. D Inf. Commun. Secur., 41, IOS Press, Amsterdam, 2015, 15–46.

• 2015-1: The case for superelliptic curvesL. Beshaj, T. Shaska, E. Zhupa,
  NATO Sci. Peace Secur. Ser. D Inf. Commun. Secur., 41, IOS Press, Amsterdam, 2015, 1–14.

• 2014-2: Cyclic curves over the reals, M. Izquierdo, T. Shaska, 
  NATO Sci. Peace Secur. Ser. D Inf. Commun. Secur., 41, IOS Press, Amsterdam, 2015, 70–83.

• 2014-1: Heights on algebraic curves, T. Shaska, L. Beshaj,
  NATO Sci. Peace Secur. Ser. D Inf. Commun. Secur., 41, IOS Press, Amsterdam, 2015, 137–175.

• 2013-7: Bielliptic curves of genus 3 in hyperelliptic moduli,
  Appl. Algebra Engrg. Comm. Comput. 24  387--412 (2013) https://doi.org/10.1007/s00200-013-0209-9 

• 2013-4: 2-Weierstrass points of genus 3 hyperelliptic curves with extra involutions,
  Comm. Algebra, 45  1879--1892  (2017). https://doi.org/10.1080/00927872.2016.1226861   

• 2013-2: Theta functions and symmetric weight enumerators for codes over imaginary quadratic fields, T. Shaska, C. Shor,
  Designs, Codes and Cryptography  76  217--235  (2015)   https://doi.org/10.1007/s10623-014-9943-7 

• 2013-1: On Jacobians of curves with superelliptic components, L. Beshaj, T. Shaska, C. Shor,
  Contemp. Math., 629, American Mathematical Society, Providence, RI, 2014, 1–14.   

• 2013-i: Computational algebraic geometry and its applications, T. Shaska,
  Appl. Algebra Engrg. Comm. Comput.  24  309--311  (2013)   https://doi.org/10.1007/s00200-013-0204-1

• 2013-ii: Computational algebraic geometry, T. Shaska,
  J. Symbolic Comput.  57  1--2  (2013)   https://doi.org/10.1016/j.jsc.2013.05.001

• 2012-2:  Genus two curves with many elliptic subcovers, T. Shaska;
  Comm. Algebra,  44  4450--4466  (2016)  https://doi.org/10.1080/00927872.2015.1027365

• 2012-1: Some remarks on the hyperelliptic moduli of genus 3, T. Shaska;
  Comm. Algebra,   42  4110--4130  (2014)  https://doi.org/10.1080/00927872.2013.791305   

• 2011-2: On superelliptic curves of level $n$ and their quotients, L. Beshaj, V. Hoxha, T. Shaska,
  Albanian J. Math.  5  115--137  (2011)

• 2011-1: Quantum codes from superelliptic curves, A. Elezi, T. Shaska,
  Albanian J. Math.  5  175--191  (2011)

• 2010-1: The arithmetic of genus two curves, T. Shaska, L. Beshaj, 
  NATO Sci. Peace Secur. Ser. D Inf. Commun. Secur., 29, IOS Press, Amsterdam, 2011, 59–98.

• 2009-1: Theta functions and algebraic curves with automorphisms, T. Shaska, S. Wijesiri, 
  NATO Sci. Peace Secur. Ser. D Inf. Commun. Secur., 24, IOS Press, Amsterdam, 2009, 193–237.

• 2008-5: On some applications of graphs to cryptography and turbocoding, T. Shaska, V. Ustimenko; 
  Albanian J. Math.  2  249--255  (2008)

• 2008-4: Degree even coverings of elliptic curves by genus 2 curves, N. Pjero, M. Ramasaco, T. Shaska; 
  Albanian J. Math.  2  241--248  (2008)

• 2008-3: Determining equations of families of cyclic curves, R. Sanjeewa, T. Shaska, 
  Albanian J. Math.  2  199--213  (2008)

• 2008-2: On the homogeneous algebraic graphs of large girth and their applications, T. Shaska, V. Ustimenko;
  Linear Algebra Appl.  430  1826--1837  (2009)    https://doi.org/10.1016/j.laa.2008.08.023

• 2008-1: Degree 4 coverings of elliptic curves by genus 2 curves, T. Shaska, S. Wijesiri, S. Wolf, L. Woodland,
  Albanian J. Math.  2  307--318  (2008)

• 2007-5: Quantum Codes from Algebraic Curves with Automorphisms,
  Condensed Matter Physics 2008, Vol. 11, No 2(54), pp. 383–396

• 2007-4: Some open problems in computational algebraic geometry, T. Shaska;
  Albanian J. Math.  1  297--319  (2007)

• 2007-3: Thetanulls of cyclic curves of small genus, E. Previato, T. Shaska, S. Wijesiri; 
  Albanian J. Math.  1  253--270  (2007)

• 2007-2: Codes over rings of size {$p^2$} and lattices over imaginary quadratic fields, T. Shaska, C. Shor, S. Wijesiri,
  Finite Fields Appl.  16  75--87  (2010)   https://doi.org/10.1016/j.ffa.2010.01.005 

• 2006-4: Subvarieties of the hyperelliptic moduli determined by group actions,
  Serdica Math. J.  32  355--374  (2006)

• 2006-3: Codes over F_{p^2} and F_pxF_p, lattices, and theta functions, T. Shaska, C. Shor, 
  Ser. Coding Theory Cryptol., 3, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2007, 70–80. 

• 2006-2: Codes over rings of size four, Hermitian lattices, and corresponding theta functions, T. Shaska, S. Wijesiri,
  Proc. Amer. Math. Soc.  136  849--857  (2008)    https://doi.org/10.1090/S0002-9939-07-09152-6   

• 2006-1: On the automorphism groups of some AG-codes based on $C_{a,b}$ curves, T. Shaska, Q. Wang,
  Serdica J. Comput.  1  193--206  (2007)

• 2005-4: A Maple package for hyperelliptic curves, T. Shaska, S. Zheng, 399--408  (2005)

• 2005-3: Hyperelliptic curves of genus 3 with prescribed automorphism group, J. Gutierrez, D. Sevilla, T. Shaska, 
  Lecture Notes Ser. Comput., 13, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2005, 109–123. 

• 2005-2: Hyperelliptic curves with reduced automorphism group $A_5$, D. Sevilla, T. Shaska,
  Appl. Algebra Engrg. Comm. Comput.  18  3--20  (2007)    https://doi.org/10.1007/s00200-006-0030-9 

• 2005-1: Genus 2 curves that admit a degree 5 map to an elliptic curve, K. Magaard, T. Shaska, H. Volklein,
  Forum Math.  21  547--566  (2009)   https://doi.org/10.1515/FORUM.2009.027   

• 2004-3: Invariants of binary forms, V. Krishnamoorthy, T. Shaska, H. Volklein,
  Dev. Math., 12, Springer, New York, 2005, 101–122. 

• 2004-2:  Genus two curves covering elliptic curves: a computational approach,
  Lecture Notes Ser. Comput., 13, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2005, 206–231. 

• 2004-1: Galois groups of prime degree polynomials with nonreal roots, A. Bialostocki, T. Shaska,
  Lecture Notes Ser. Comput., 13, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2005, 243–255. 

• 2003-4: Computational algebra and algebraic curves,
  SIGSAM Bull. 37, 117--124  (2003)  https://doi.org/10.1145/968708.968713   

• 2003-3: Hyperelliptic curves with extra involutions, J. Gutierrez, T. Shaska, 
  LMS J. Comput. Math.  8  102--115  (2005)  https://doi.org/10.1112/S1461157000000917  

• 2003-2: Some special families of hyperelliptic curves, T. Shaska,
  J. Algebra Appl.  3  75--89  (2004) https://doi.org/10.1142/S0219498804000745  

• 2003-1: On the generic curve of genus 3, T. Shaska, J. Thompson,
  Contemp. Math., Contemporary Math., 369, AMS,  Providence, RI, 2005 

• 2002-3: Determining the automorphism group of a hyperelliptic curve,  (ISSAC 2003)
  Proceedings of the 2003 International Symposium on Symbolic and Algebraic Computation, 248–254. (ACM), 2003

• 2002-2: Computational aspects of hyperelliptic curves,
  Lecture Notes Ser. Comput., 10, World Scientific Publishing Co., Inc., River Edge, NJ, 2003, 248–257. 

• 2002-1: Genus 2 curves with (3,3)-split Jacobian and large automorphism group, (ANTS 2003)
  Lecture Notes in Comput. Sci., 2369, Springer-Verlag, Berlin, 2002, 205–218.   math/0201008

• 2001-2: The locus of curves with prescribed automorphism group, K. Magaard, T. Shaska, S. Shpectorov, H. Volklein,
  Sürikaisekikenkyüsho K\B{o}kyüroku 112--141  (2002) math/0205314

• 2001-1: Genus 2 fields with degree 3 elliptic subfields,
  Forum Math. 16  263--280  (2004)  https://doi.org/10.1515/form.2004.013  math/0109155

• 2001-0: Curves of genus two covering elliptic curves, 
  Thesis (Ph.D.)--University of Florida pg.72 (2001).

• 2000-2: Elliptic subfields and automorphisms of genus 2 function fields,  T. Shaska, H. Volklein;
  Algebra, arithmetic and geometry with applications (Abhyankar's 70th birthday), West Lafayette, IN, 2000, 703–723, Springer-Verlag, Berlin, (2004)  math/0107142 

• 2000-1: Curves of genus 2 with (n,n)-decomposable Jacobians,  
  J. Symbolic Comput. 31,  603--617, (2001). https://doi.org/10.1006/jsco.2001.0439   math/0312285