Upgraded methods for the effective computation of marked schemes on a strongly stable ideal

Let $J\subset S=K[x_0,...,x_n]$ be a monomial strongly stable ideal. The collection \Mf(J) of the homogeneous polynomial ideals $I$, such that the monomials outside $J$ form a $K$-vector basis of $S/I$, is called a {\em $J$-marked family}. It can be endowed with a structure of affine scheme, called a {\em $J$-marked scheme}. For special ideals $J$, $J$-marked schemes provide an open cover of the Hilbert scheme \hilbp, where $p(t)$ is the Hilbert polynomial of $S/J$. Those ideals more suitable to this aim are the $m$-truncation ideals $\underline{J}_{\geq m}$ generated by the monomials of degree $\geq m$ in a saturated strongly stable monomial ideal $\underline{J}$. Exploiting a characterization of the ideals in \Mf(\underline{J}_{\geq m}) in terms of a Buchberger-like criterion, we compute the equations defining the $\underline{J}_{\geq m}$-marked scheme by a new reduction relation, called {\em superminimal reduction}, and obtain an embedding of \Mf(\underline{J}_{\geq m}) in an affine space of low dimension. In this setting, explicit computations are achievable in many non-trivial cases. Moreover, for every $m$, we give a closed embedding \phi_m: \Mf(\underline{J}_{\geq m})\hookrightarrow \Mf(\underline{J}_{\geq m+1}), characterize those $\phi_m$ that are isomorphisms in terms of the monomial basis of $\underline{J}$, especially we characterize the minimum integer $m_0$ such that $\phi_m$ is an isomorphism for every $m\geq m_0$.Comment: 28 pages; this paper contains and extends the second part of the paper posed at arXiv:0909.2184v2[math.AG]; sections are now reorganized and the general presentation of the paper is improved. Final version accepted for publicatio

Mini-Workshop: Formal Methods in Commutative Algebra: A View Toward Constructive Homological Algebra

The purpose of the mini-workshop is to bring into the same place different mathematical communities that study constructive homological algebra and are motivated by different applications (e.g., constructive algebra, symbolic computation, proof theory, algebraic topology, mathematical systems theory, D-modules, dynamical systems theory) so that they can share their results, techniques, softwares and experiences. Through the development of a unified terminology, common mathematical problems, which naturally appear when making homological algebra constructive, were discussed

Decompositions of commutative monoid congruences and binomial ideals

Primary decomposition of commutative monoid congruences is insensitive to certain features of primary decomposition in commutative rings. These features are captured by the more refined theory of mesoprimary decomposition of congruences, introduced here complete with witnesses and associated prime objects. The combinatorial theory of mesoprimary decomposition lifts to arbitrary binomial ideals in monoid algebras. The resulting binomial mesoprimary decomposition is a new type of intersection decomposition for binomial ideals that enjoys computational efficiency and independence from ground field hypotheses. Binomial primary decompositions are easily recovered from mesoprimary decomposition.Comment: 62 pages, 7 figures, v2: small improvements over v1, v3: added Problem 17.7, Corollary 4.15 and other small refinements, v4: major revision: definition of witness adjusted (Section 4), and incorrect claims concerning binomial irreducible decomposition excised (Section 15), v5: various corrections and improvements. Final version, accepted by Algebra and Number Theor

Algebraic Statistics

Algebraic Statistics is concerned with the interplay of techniques from commutative algebra, combinatorics, (real) algebraic geometry, and related fields with problems arising in statistics and data science. This workshop was the first at Oberwolfach dedicated to this emerging subject area. The participants highlighted recent achievements in this field, explored exciting new applications, and mapped out future directions for research