Álgebra comutativa em quatro movimentos, with Eduardo Tengan. Coleção Projeto Euclides, IMPA, 2nd edition, 2020.
Uma Introdução à Álgebra Abstrata. Independent book project, 2026.
An independent and evolving work born from lecture notes for Algebra I, II, and III, courses taught at ICMC–USP between 2010 and 2025. Shaped by fifteen years of teaching, by questions and contributions from students, by exchanges with colleagues, and by a dialogue with classical texts, the book offers a unified path through groups, rings, fields, and Galois theory.
R. Alvarenga, H. Borges, On the number of elements with prescribed norm and trace. Preprint (2023). arXiv.
H. Borges, L. Reis. Minimal value set polynomials. Advances in Mathematics 482 (2025), 110627. DOI.
H. Borges, J. P. Guardieiro, C. Salgado, J. Top. Tate-Shafarevich Results for Quartic Twists in Characteristic 2. Journal of Algebra and Its Applications 24 (2025), 2541025. DOI.
H. Borges, C. Gonçalves. The p-rank of curves of Fermat type. Finite Fields and Their Applications 97 (2024), 102430. DOI.
H. Borges, G. Korchmáros, P. Speziali. Plane curves with a large linear automorphism group in characteristic p. Finite Fields and Their Applications 96 (2024), 102402. DOI.
H. Borges, J. A. Oliveira, F. E. Brochero Martínez, On the number of rational points on Artin-Schreier hypersurfaces. Finite Fields and Their Applications, 90 (2023), 102229
H. Borges, S. Fukasawa, An elementary abelian p-cover of the Hermitian curve with many automorphisms, Mathematische Zeitschrift, 302, (2022) 695–706.
H. Borges, G. Duran. Weierstrass Pure Gaps on Curves with Three Distinguished Points. IEEE Transactions on Information Theory 68 (2022). DOI.
H. Borges, L. Reis. Minimal value set polynomials over fields of size p³. Proceedings of the American Mathematical Society 149 (2021), 3639–3649. DOI.
D. Bartoli, H. Borges, L. Quoos. Rational functions with small value sets. Journal of Algebra, v. 565, p. 675-690, 2021.
N. Arakelian, H. Borges, P. Speziali, The Hurwitz curve over a finite field and its Weierstrass points for the morphism of lines. Finite Fields and Their Applications, v. 73, p. 101842, 2021.
H. Borges, S. Fukasawa. Galois Points for double-Frobenius Nonclassical Curves. Finite Fields and Their applications, v. 61, p. 101579, 2020.
H. Borges, M. Coutinho, On some generalized Fermat curves and chords of an affinely regular polygon inscribed in a hyperbola. Journal of Pure and Applied Algebra 224 (2020) 239-249.
M. Abdon, H. Borges, L. Quoos. Weierstrass points on Kummer extensions. Advances in Geometry 19 (2019) 323-333.
H. Borges, A. Sepúlveda, G. Tizziotti. Subcovers and Codes on a Class of Trace-Defining Curves. IEEE Transactions on Information Theory 65 (2019), 2101–2106. DOI.
N. Arakelian, H. Borges. Bounds for the number of points on curves over finite fields. Israel Journal of Mathematics 228 (2018), 177–199. DOI.
H. Borges, G. Cook and M. Coutinho. Plane sections of Fermat Surfaces over Finite Fields. Finite Fields and their Applications 52 (2018), p. 156-173.
H. Borges, R. Conceição. A new family of Castle and Frobenius nonclassical curves. Journal of Pure and Applied Algebra 222 (2018) 994-1002.
H. Borges, M. Homma. Points on singular Frobenius nonclassical curves. Bulletin of the Brazilian Mathematical Society 48 (2017) 93-101 PDF
N. Arakelian, H. Borges. Frobenius nonclassicality of Fermat curves with respect to cubics. Israel Journal of Mathematics 218 (2017) 273-297.
H. Borges. Frobenius nonclassical components of curves with separated variables. Journal of Number Theory, v. 159, 402–425 (2016).
H. Borges, A. Sepúlveda, G. Tizziotti. Weierstrass semigroup and Automorphism group of the curves Xn,r. Finite Fields Appl. 36, 121–132 (2015).
N. Arakelian, H. Borges. Frobenius nonclassicality with respect to linear systems of curves of arbitrary degree. Acta Arithmetica, v.167, p.43-66, (2015).
H. Borges, F. Torres, B. Motta. Complete Arcs arising from a generalization of the Hermitian Curve. Acta Arithmetica, v.164, p.101-118, (2014).
H. Borges, R. Conceição. On the characterization of minimal value set polynomials. Journal of Number Theory 133 , no. 6, 2021– 2035 (2013).
H. Borges, B. Motta, F. Torres. Complete arcs arising from a generalization of the Hermitian curve (extended abstract). Electronic Notes in Discrete Mathematics 40 (2013), 271–275. DOI.
H. M. Borges Filho. On multi-Frobenius nonclassical plane curves. Archiv der Mathematik v. 93, p. 541-553, (2009).
H. Borges. On complete (N, d)-arcs derived from plane curves. Finite Fields Appl. 15, 82–96 (2009).
H. Borges, M. Coutinho, On the Zeta function and the automorphism group of the generalized Suzuki curve. Transactions of the American Mathematical Society, 374, (2021), 1899–1917.