查看更多>>摘要：We present a certified algorithm based on subdivision for computing an isotopic approximation to any number of algebraic curves in the plane. Our algorithm is based on the certified curve approximation algorithm of Plantinga and Vegter. The main challenge in this algorithm is to correctly and efficiently identify and isolate all intersections between the curves. To overcome this challenge, we introduce a new and simple test that guarantees the global correctness of our output. A main step in our algorithm for approximating any number of curves is to correctly approximate a pair of curves. In addition to developing the details of this special case, we provide complexity analyses for both the number of steps and the bit-complexity of this algorithm using both worst-case bounds as well as those based on continuous amortization and condition numbers. (c) 2025 Elsevier Ltd. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
查看更多>>摘要：Let K be a field and P = K[x1, ... , xn]. The technique of elimination by substitution is based on discovering a coherently Z = (z1,..., zs)-separating tuple of polynomials ( f1,..., fs) in an ideal I, i.e., on finding polynomials such that fi= zi - hi with hi is an element of K[X\Z]. Here we elaborate on this technique in the case when Pis nonnegatively graded. The existence of a coherently Z-separating tuple is reduced to solving several P0-module membership problems. Best separable re-embeddings, i.e., isomorphisms P/I --> K[X\Z]/ (I boolean AND K[X\Z]) with maximal #Z, are found degree-by-degree. They turn out to yield optimal re-embeddings in the positively graded case. Viewing P0 --> P/I as a fibration over an affine space, we show that its fibers allow optimal Z-separating re-embeddings, and we provide a criterion for a fiber to be isomorphic to an affine space. In the last section we introduce a new technique based on the solution of a unimodular matrix problem which enables us to construct automorphisms of P such that additional Z-separating re-embeddings are possible. One of the main outcomes is an algorithm which allows us to explicitly compute a homogeneous isomorphism between P/I and a non-negatively graded polynomial ring if P/I is regular. (c) 2025 The Author(s). Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http:// creativecommons.org/licenses/by-nc-nd/4.0/).
查看更多>>摘要：Computing the arrangement of quadrics in 3D is a fundamental problem in symbolic computation, with challenges arising when handling degenerate cases and asymptotic critical values. Stateof-the-art methods typically require a generic change of coordinates to manage these asymptotes, rendering certain problems intractable. A specific instance of this challenge appears in digital geometry, where comparing 3D shapes up to isometry requires applying a 3D rigid motion on Z3 and mapping the result back to Z3, a process typically achieved via a digitization operator. However, such motions do not preserve the topology of digital objects, making the analysis of digitized rigid motions crucial. Our main contribution is the decomposition of the 6D parameter space of digitized rigid motions for image patches of radius up to three. This problem reduces to computing the arrangement of up to 741 quadrics, some of which are degenerate. To address the computational challenges, we introduce and implement a new algorithm for computing arrangements of quadrics in 3D, specifically designed to handle degenerate directions and asymptotic critical values. This approach allows us to overcome the limitations of existing methods, making the problem tractable in the context of digital geometry. (c) 2025 Elsevier Ltd. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
查看更多>>摘要：We present effective procedures to calculate regular normal cones and other related objects using quantifier elimination. This method of normal cone calculations is complementary to computing Lagrangians and it works best at points where the constraint qualifications fail and extra work for other methods becomes inevitable. This method also serves as a tool to calculate the regular co-derivative for semismooth* Newton methods. We list algorithms and their demonstrations of different use cases for this approach. (c) 2025 Published by Elsevier Ltd.
查看更多>>摘要：How to handle division in systems that compute with logical formulas involving what would otherwise be polynomial constraints over the real numbers is a surprisingly difficult question. This paper argues that existing approaches from both the computer algebra and computational logic communities are unsatisfactory for systems that consider the satisfiability of formulas with quantifiers or that perform quantifier elimination. To address this, we propose the notion of the fair-satisfiability of a formula, use it to characterize formulas with divisions that are well-defined, meaning that they adequately guard divisions against division by zero, and provide a translation algorithm that converts a formula with divisions into a purely polynomial formula that is satisfiable if and only if the original formula is fair-satisfiable. This provides a semantics for division with some nice properties, which we describe and prove in the paper. Published by Elsevier Ltd.
查看更多>>摘要：Based on a theorem by Vasconcelos, we give an algorithm for equidimensional decomposition of algebraic sets using syzygy computations via Gr & ouml;bner bases. This algorithm avoids the use of elimination, homological algebra and processing the input equations one-by-one present in previous algorithms. We experimentally demonstrate the practical interest of our algorithm compared to the state of the art. (c) 2025 Elsevier Ltd. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
查看更多>>摘要：The classical Chow form encodes any projective variety by one equation. We here introduce the Chow-Lam form for subvarieties of a Grassmannian. By evaluating the Chow-Lam form at twistor coordinates, we obtain universal projection formulas. These were pioneered by Thomas Lam for positroid varieties in the study of amplituhedra, and we develop his approach further. Universal formulas for branch loci are obtained from Hurwitz-Lam forms. Our focus is on computations and applications in geometry. (c) 2025 The Author(s). Published by Elsevier Ltd. This is an open access article under the CC BY license (http:// creativecommons.org/licenses/by/4.0/).
查看更多>>摘要：We study the problem of characterizing linear preserver subgroups of algebraic varieties, with a particular emphasis on secant varieties and other varieties of tensors. We introduce a number of techniques built on different geometric properties of the varieties of interest. Our main result is a simple characterization of the linear preservers of secant varieties of Segre varieties in many cases, including sigma r((Pn-1)xk) for all r 2J. We also characterize the linear preservers of several other sets of tensors, including subspace varieties, the variety of slice rank one tensors, symmetric tensors of bounded Waring rank, the variety of biseparable tensors, and hyperdeterminantal surfaces. Computational techniques and applications in quantum information theory are discussed. We provide geometric proofs for several previously known results on linear preservers.<br /> (c) 2025 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY license (http:// creativecommons.org/licenses/by/4.0/).
查看更多>>摘要：In this article, we describe primitive quandles with the help of primitive permutation groups. As a consequence, we enumerate finite non-affine primitive quandles up to order 4096. (c) 2025 Elsevier Ltd. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
查看更多>>摘要：We embed several copies of the derived category of a quiver and certain line bundles in the derived category of an associated moduli space of representations, giving the start of a semiorthogonal decomposition. This mirrors the semiorthogonal decompositions of moduli of vector bundles on curves. Our results are obtained with QUIVERTOOLS, an open-source package of tools for quiver representations, their moduli spaces and their geometrical properties. (c) 2025 Elsevier Ltd. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
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