Elías García, Joan2024-07-112024-07-112022-06-251660-5446https://hdl.handle.net/2445/214510The aim of this note is to exploit a new relationship between additive combinatorics and the geometry of monomial projective curves. We associate to a finite set of non-negative integers $A=\left\{a_1, \ldots, a_n\right\}$ a monomial projective curve $C_A \subset \mathbb{P}_{\mathbf{k}}^{n-1}$ such that the Hilbert function of $C_A$ and the cardinalities of $s A:=\left\{a_{i_1}+\cdots+a_{i_s} \mid 1 \leq i_1 \leq \cdots \leq i_s \leq n\right\}$ agree. The singularities of $C_A$ determines the asymptotic behaviour of $|s A|$, equivalently the Hilbert polynomial of $C_A$, and the asymptotic structure of $S A$. We show that some additive inverse problems can be translate to the rigidity of Hilbert polynomials and we improve an upper bound of the Castelnuovo-Mumford regularity of monomial projective curves by using results of additive combinatorics.11 p.application/pdfengcc-by (c) Joan Elias García, 2022http://creativecommons.org/licenses/by/3.0/es/Àlgebra commutativaSuccessions (Matemàtica)Corbes algebraiquesCommutative algebraSequences (Mathematics)Algebraic curvesinfo:eu-repo/semantics/article7493752024-07-11info:eu-repo/semantics/openAccess