An official website of the United States government
Symmetric strongly shifted ideals constitute a class of monomial ideals which are equipped with an action of the symmetric group and are analogous to the well-studied class of strongly stable monomial ideals. In this paper, we focus on algebraic and combinatorial properties of symmetric strongly shifted ideals. On the algebraic side, we elucidate properties that pertain to behavior under ideal operations, primary decomposition, and the structure of their Rees algebra. On the combinatorial side, we develop a notion of partition Borel generators which leads to connections to discrete polymatroids, convex polytopes, and permutohedral toric varieties.
more »
« less
Virtual resolutions of monomial ideals on toric varieties
We use cellular resolutions of monomial ideals to prove an analog of Hilbert’s syzygy theorem for virtual resolutions of monomial ideals on smooth toric varieties.
more »
« less
- Award ID(s):
- 1745638
- PAR ID:
- 10279624
- Date Published:
- Journal Name:
- Proceedings of the American Mathematical Society, Series B
- Volume:
- 8
- Issue:
- 9
- ISSN:
- 2330-1511
- Page Range / eLocation ID:
- 100 to 111
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
Principal symmetric ideals were recently introduced by Harada et al. in [The minimal free resolution of a general principal symmetric ideal, preprint (2023), arXiv:2308.03141], where their homological properties are elucidated. They are ideals generated by the orbit of a single polynomial under permutations of variables in a polynomial ring. In this paper, we determine when a product of two principal symmetric ideals is principal symmetric and when the powers of a principal symmetric ideal are again principal symmetric ideals. We characterize the ideals that have the latter property as being generated by polynomials invariant up to a scalar multiple under permutation of variables. Recognizing principal symmetric ideals is an open question for the purpose of which we produce certain obstructions. We also demonstrate that the Hilbert functions of symmetric monomial ideals are not all given by symmetric monomial ideals, in contrast to the non-symmetric case.more » « less
-
Abstract We study when blowup algebras are ‐split or strongly ‐regular. Our main focus is on algebras given by symbolic and ordinary powers of ideals of minors of a generic matrix, a symmetric matrix, and a Hankel matrix. We also study ideals of Pfaffians of a skew‐symmetric matrix. We use these results to obtain bounds on the degrees of the defining equations for these algebras. We also prove that the limit of the normalized regularity of the symbolic powers of these ideals exists and that their depth stabilizes. Finally, we show that, for determinantal ideals, there exists a monomial order for which taking initial ideals commutes with taking symbolic powers. To obtain these results, we develop the notion of ‐split filtrations and symbolic ‐split ideals.more » « less
-
A new approach for Gröbner bases conversion of polynomial ideals (over a field) of arbitrary dimension is presented. In contrast to the only other approach based on Gröbner fan and Gröbner walk for positive dimensional ideals, the proposed approach is simpler to understand, prove, and implement. It is based on defining for a given polynomial, a truncated sub-polynomial consisting of all monomials that can possibly become the leading monomial with respect to the target ordering: the monomials between the leading monomial of the target ordering and the leading monomial of the initial ordering. The main ingredient of the new algorithm is the computation of a Gröbner basiswith respect tothe target ordering for the ideal generated by such truncated parts of the input Gröbner basis. This is done using the extended Buchberger algorithm that also outputs the relationship between the input and output bases. That information is used in attempts to recover a Gröbner basis of the idealwith respect tothe target ordering. In general, more than one iteration may be needed to get a Gröbner basiswith respect tothe target ordering since truncated polynomials may miss some leading monomials. The new algorithm has been implemented inMapleand its operation is illustrated using an example. The performance of this implementation is compared with the implementations of other approaches inMaple.In practice, a Gröbner basiswith respect toa target ordering can be computed in a single iteration on most examples.more » « less
-
We prove that the multiplicity of a filtration of a local ring satisfies various convexity properties. In particular, we show the multiplicity is convex along geodesics. As a consequence, we prove that the volume of a valuation is log convex on simplices of quasi-monomial valuations and give a new proof of a theorem of Xu and Zhuang on the uniqueness of normalized volume minimizers. In another direction, we generalize a theorem of Rees on multiplicities of ideals to filtrations and characterize when the Minkowski inequality for filtrations is an equality under mild assumptions.more » « less
- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1090/bproc/72
