arXiv subjects

Daniel Qin

Publications and source records attributed to Daniel Qin.

4 recordsLinked to original sources

arXiv2026

An algorithm for invariants of elementary abelian groups

When we consider a finite abelian group acting linearly on a polynomial ring, we can find monomial generators for the subring of invariants. By Noether's degree bound and Hilbert's finiteness theorem, we know that there are finitely many minimal generators, but efficiently finding a generating set is not a trivial task. We present a new algorithm for computing the invariant ring for elementary abelian groups acting on polynomial rings with complex coefficients (or any other field of characteristic zero). We follow a two-step process: first we generate a collection of $n-k$ "seed" invariants by calculating the kernel of a weight matrix that encodes our action. After we find the seeds, we "grow" them into a generating set for the invariant ring by exploiting the lattice structure of invariants modulo $p$. Our algorithm performs better than the one currently available in Macaulay2, allowing us to compute invariants more quickly in this setting.

math.AC↗

arXiv2026

Induced Lorentzian and volume polynomials

Suppose one has a party of $m$ people, whose expertise collectively covers $n$ topics. Given a subset $T$ of the topics, one wishes to form a panel of $|T|$ people from the party such that $T$ can be covered by assigning a distinct topic to each panel member with the expertise. We show that the numbers of such panels, as $T$ varies, form a Lorentzian polynomial. We achieve this by showing that a certain linear operator on polynomials, which we call the ``inducing operator'' for its connection to induced (poly)matroids, preserves Lorentzian polynomials and realizable volume polynomials.

math.CO↗

arXiv2025

Stem-Symmetry, Comb Products, and their Relation to Amoeba Graphs

Local and global amoebas are families of labeled graphs that satisfy interpolation properties on a fixed vertex set. A labeled graph $G$ on $n$ vertices is a local amoeba (resp. global amoeba) if there exists a sequence of feasible edge-replacements between any two labelled embeddings of $G$ into $K_n$ (resp. $K_{n+1}$). Here, a feasible edge-replacement removes an edge and reinserts it so that the resulting graph is isomorphic to $G$; the induced relabeling yields a class of permutations of the label set. Motivated by classical group theoretic ideas, we introduce the hang group, a new invariant that can encode how local amoebas embed into larger ones. Using this framework, we identify necessary and sufficient conditions connecting stem-symmetric and hang-symmetric graphs with local and global amoebas. In particular, we show how hang-symmetry and stem-symmetry conditions propagate under the addition of leaves and isolated vertices, in turn yielding constructive criteria for both local and global amoebas. Finally, via wreath products, we provide four sets of sufficient conditions, one for each property, guaranteeing when the comb product is a local amoeba, a global amoeba, stem-symmetric, or hang-symmetric. These results strengthen and generalize existing constructions of local and global amoebas.

math.CO↗

arXiv2025

Symmetric Lorentzian Polynomials

We study the class of Lorentzian symmetric polynomials and Lorentzian symmetric functions, which are defined to be symmetric functions for which every truncation of variables is Lorentzian. Similar to the space of Lorentzian polynomials, we show that the space of Lorentzian symmetric polynomials is homeomorphic to a closed Euclidean ball. Our main result is a reduction scheme that significantly reduces the complexity of testing for Lorentzianity. Using this method, we provide explicit semialgebraic descriptions of the spaces of Lorentzian symmetric polynomials and functions for degrees up to six. These techniques can also be applied to simplify the proofs to known cases of Lorentzian symmetric functions. We conclude by showing that some natural symmetric operators fail to preserve Lorentzianity which in turn highlights an inherent tension between symmetry in variables and the Lorentzian property.

math.CO↗