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1. -SIGNATURE UNDER BIRATIONAL MORPHISMS

   https://doi.org/10.1017/fms.2019.6  

   MA, LINQUAN; POLSTRA, THOMAS; SCHWEDE, KARL; TUCKER, KEVIN (January 2019, Forum of Mathematics, Sigma)

   We study $$F$$ -signature under proper birational morphisms $$\unicode[STIX]{x1D70B}:Y\rightarrow X$$ , showing that $$F$$ -signature strictly increases for small morphisms or if $$K_{Y}\leqslant \unicode[STIX]{x1D70B}^{\ast }K_{X}$$ . In certain cases, we can even show that the $$F$$ -signature of $$Y$$ is at least twice as that of  $$X$$ . We also provide examples of $$F$$ -signature dropping and Hilbert–Kunz multiplicity increasing under birational maps without these hypotheses. 

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2. CONTINUITY OF HILBERT–KUNZ MULTIPLICITY AND F-SIGNATURE

   https://doi.org/10.1017/nmj.2018.43  

   POLSTRA, THOMAS; SMIRNOV, ILYA (September 2020, Nagoya Mathematical Journal)

   We establish the continuity of Hilbert–Kunz multiplicity and F-signature as functions from a Cohen–Macaulay local ring $$(R,\mathfrak{m},k)$$ of prime characteristic to the real numbers at reduced parameter elements with respect to the $$\mathfrak{m}$$ -adic topology. 

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3. A Volume = Multiplicity formula for p-families of ideals

   https://doi.org/10.1090/proc/16451  

   Das, Sudipta (October 2023, Proceedings of the American Mathematical Society)

   In this article, we work with certain families of ideals called $p$-families in rings of prime characteristic. This family of ideals is present in the theories of tight closure, Hilbert-Kunz multiplicity, and $F$-signature. For each $p$-family of ideals, we attach an Euclidean object called $p$-body, which is analogous to the Newton Okounkov body associated with a graded family of ideals. Using the combinatorial properties of $p$-bodies and algebraic properties of the Hilbert-Kunz multiplicity, we establish a Volume = Multiplicity formula for $p$-families of ${\mathfrak{m}}_R$-primary ideals in a Noetherian local ring $R$. 

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4. Punctual Hilbert scheme and certified approximate singularities

   https://doi.org/10.1145/3373207.3404024  

   Mantzaflaris, Angelos; Mourrain, Bernard; Szanto, Agnes (July 2020, ISSAC '20: Proceedings of the 45th International Symposium on Symbolic and Algebraic Computation)

   In this paper we provide a new method to certify that a nearby polynomial system has a singular isolated root and we compute its multiplicity structure. More precisely, given a polynomial system f = (f1, ..., fN) ∈ C[x1, ..., xn]^N, we present a Newton iteration on an extended deflated system that locally converges, under regularity conditions, to a small deformation of f such that this deformed system has an exact singular root. The iteration simultaneously converges to the coordinates of the singular root and the coefficients of the so-called inverse system that describes the multiplicity structure at the root. We use α-theory test to certify the quadratic convergence, and to give bounds on the size of the deformation and on the approximation error. The approach relies on an analysis of the punctual Hilbert scheme, for which we provide a new description. We show in particular that some of its strata can be rationally parametrized and exploit these parametrizations in the certification. We show in numerical experimentation how the approximate inverse system can be computed as a starting point of the Newton iterations and the fast numerical convergence to the singular root with its multiplicity structure, certified by our criteria. 

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5. NUMERICAL SEMIGROUPS FROM RATIONAL MATRICES II: MATRICIAL DIMENSION DOES NOT EXCEED MULTIPLICITY

   https://doi.org/10.1017/S0004972724001035  

   CHHABRA, ARSH; GARCIA, STEPHAN RAMON; O’NEILL, CHRISTOPHER (December 2024, Bulletin of the Australian Mathematical Society)

   Abstract We continue our study of exponent semigroups of rational matrices. Our main result is that the matricial dimension of a numerical semigroup is at most its multiplicity (the least generator), greatly improving upon the previous upper bound (the conductor). For many numerical semigroups, including all symmetric numerical semigroups, our upper bound is tight. Our construction uses combinatorially structured matrices and is parametrised by Kunz coordinates, which are central to enumerative problems in the study of numerical semigroups. 

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