Research | alca

RESEARCH INTERESTS

My research interests are in commutative algebra and its applications to the related fields of algebraic geometry and cryptography.

Commutative algebra: numerical functions and multiplicities of rings of positive characteristic (Hilbert-Kunz, generalized Hilbert-Kunz, F-signature), symmetric signature, invariant rings.
Algebraic geometry: arithmetical rank, Gale transform, vector bundles over curves, point configurations on projective curves.
Cryptography: algorithms for polynomial systems solving and related invariants (solving degree, last fall degree), multivariate cryptography, history of cryptography.

PREPRINTS

Simplices osculating rational normal curves (with Enrico Carlini, Luca Schaffler)

preprint, [arXiv:2404.03922]

F-signature functions of diagonal hypersurfaces (with Samuel Shideler, Kevin Tucker, Francesco Zerman)

preprint, [arXiv:2403.12863]

Structure of CSS and CSS-T Quantum Codes (with Elena Berardini, Alberto Ravagnani)

preprint, [arXiv:2310.16504]

PUBLICATIONS

Multidegrees, prime ideals, and non-standard gradings (with Yairon Cid-Ruiz, Aldo Conca)

Advances in Mathematics, Vol. 435, Part A, 2023.

[journal], [arXiv:2208.07238]

Determinantal varieties from point configurations on hypersurfaces (with Han-Bom Moon, Luca Schaffler)
International Mathematics Research Notices, Vol. 2023, Issue 22, pp. 19743-19772.

[journal], [arXiv:2211.13177]

The complexity of solving Weil restriction systems (with Michela Ceria, Elisa Gorla)
Journal of Algebra, Vol. 621, pp. 116-133, 2023.

[journal], [arXiv:2112.10506]

Solving degree, last fall degree, and related invariants (with Elisa Gorla)
Journal of Symbolic Computation, Vol. 114, pp. 322-335, 2023.

[journal], [arXiv:2112.05579]

The complexity of MinRank (with Elisa Gorla)

WiNE III: Research Directions in Number Theory A. Cojocaru, S. Ionica and E. Lorenzo Garcia Eds., Springer, pp. 163-169, 2021.

[book chapter], [arXiv:1905.02682]

A Pascal's Theorem for rational normal curves (with Luca Schaffler)
Bulletin of the London Mathematical Society, Vol 53, No.5, pp. 1470-1485, 2021.

[journal], [arXiv:1903.00460]

Point configurations, phylogenetic trees, and dissimilarity vectors (with Noah Giansiracusa, Han-Bom Moon, Luca Schaffler)

Proceedings of the National Academy of Sciences of the United States of America (PNAS), Vol.118, No.12, 2021.

[journal], [arXiv:2005.02629]

Solving Multivariate Polynomial Systems and an Invariant from Commutative Algebra (with Elisa Gorla)

Arithmetic of Finite Fields, 8th International Workshop, J.C. Bajard and A. Topuzoglu Eds, Lecture Notes in Computer Science, 12542 LNCS, pp. 3-36, Springer, 2021

[book chapter], [arXiv:1706.06319]

Nearly Gorenstein cyclic quotient singularities (with Francesco Strazzanti)

Beiträge zur Algebra und Geometrie, Vol. 62, No. 4, pp. 857-870, 2021.

[journal], [arXiv:2006.03457]

F-signature function of quotient singularities (with Alessandro De Stefani)

Journal of Algebra, Vol. 523, pp. 311-341, 2019

[journal], [arXiv:1804.01281]

Differential symmetric signature in high dimension (with Holger Brenner)

Proceedings of the American Mathematical Society, Vol. 147, No. 10, pp. 4147-4159, 2019

[journal], [arXiv:1711.07239]

Equations for point configurations to lie on a rational normal curve (with Noah Giansiracusa, Han-Bom Moon, Luca Schaffler)

Advances in Mathematics, Vol. 340, pp. 653-683, 2018