math.AG

arXiv:2501.00657v1 Announce Type: new Abstract: Relative pose (position and orientation) estimation is an essential component of many robotics applications. Fiducial markers, such as the AprilTag visual fiducial system, yield a relative pose measurement from a single marker detection and provide a powerful tool for pose estimation. In this paper, we perform a Lie algebraic nonlinear observability analysis on a nonlinear dual quaternion system that is composed of a relative pose measurement model and a relative motion model. We prove that many common dual quaternion expressions yield Jacobian matrices with advantageous block structures and rank properties that are beneficial for analysis. We show that using a dual quaternion representation yields an observability matrix with a simple block triangular structure and satisfies the necessary full rank condition.

arXiv:2501.00563v1 Announce Type: cross Abstract: We present a new Python package called "motives", a symbolic manipulation package based on SymPy capable of handling and simplifying motivic expressions in the Grothendieck ring of Chow motives and other types of $\lambda$-rings. The package is able to manipulate and compare arbitrary expressions in $\lambda$-rings and, in particular, it contains explicit tools for manipulating motives of several types of commonly used moduli schemes and moduli stacks of decorated bundles on curves. We have applied this new tool to advance in the verification of Mozgovoy's conjectural formula for the motive of the moduli space of twisted Higgs bundles, proving that it holds in rank 2 and 3 for any curve of genus up to 18 and any twisting bundle of small degree.

arXiv:2412.17952v1 Announce Type: cross Abstract: Similarly to the global case, the local structure of a holomorphic subvariety at a given point is described by its local irreducible decomposition. Following the paradigm of numerical algebraic geometry, an algebraic subvariety at a point is represented by a numerical local irreducible decomposition comprised of a local witness set for each local irreducible component. The key requirement for obtaining a numerical local irreducible decomposition is to compute the local monodromy action of a generic linear projection at the given point, which is always well-defined on any small enough neighborhood. We characterize some of the behavior of local monodromy action of linear projection maps under analytic continuation, allowing computations to be performed beyond a local neighborhood. With this characterization, we present an algorithm to compute the local monodromy action and corresponding numerical local irreducible decomposition for algebraic varieties. The results are illustrated using several examples facilitated by an implementation in an open source software package.

arXiv:2412.17099v1 Announce Type: cross Abstract: A finite group with an integral representation has two induced canonical actions, one on polynomials and one on Laurent polynomials. Knowledge about the invariants is in either case applied in many computations by means of symmetry reduction techniques, for example in algebraic systems solving or optimization. In this article, we realize the two actions as the additive action on the symmetric algebra and the multiplicative action on the group algebra of a lattice with Weyl group symmetry. By constructing explicit equivariant isomorphisms, we draw algorithmic relations between the two, which allow the transfer and preservation of representation- and invariant-theoretic properties. Our focus lies on the multiplicative coinvariant space, which is identified with the regular representation and harmonic polynomials.

arXiv:2403.17250v3 Announce Type: replace-cross Abstract: We use machine learning to study the moduli space of genus two curves, specifically focusing on detecting whether a genus two curve has $(n, n)$-split Jacobian. Based on such techniques, we observe that there are very few rational moduli points with small weighted moduli height and $(n, n)$-split Jacobian for $n=2, 3, 5$. We computational prove that there are only 34 genus two curves (resp. 44 curves) with (2,2)-split Jacobians (resp. (3,3)-split Jacobians) and weighted moduli height $\leq 3$. We discuss different machine learning models for such applications and demonstrate the ability to detect splitting with high accuracy using only the Igusa invariants of the curve. This shows that artificial neural networks and machine learning techniques can be highly reliable for arithmetic questions in the moduli space of genus two curves and may have potential applications in isogeny-based cryptography.