88 research outputs found

    Computation and Physics in Algebraic Geometry

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    Physics provides new, tantalizing problems that we solve by developing and implementing innovative and effective geometric tools in nonlinear algebra. The techniques we employ also rely on numerical and symbolic computations performed with computer algebra. First, we study solutions to the Kadomtsev-Petviashvili equation that arise from singular curves. The Kadomtsev-Petviashvili equation is a partial differential equation describing nonlinear wave motion whose solutions can be built from an algebraic curve. Such a surprising connection established by Krichever and Shiota also led to an entirely new point of view on a classical problem in algebraic geometry known as the Schottky problem. To explore the connection with curves with at worst nodal singularities, we define the Hirota variety, which parameterizes KP solutions arising from such curves. Studying the geometry of the Hirota variety provides a new approach to the Schottky problem. We investigate it for irreducible rational nodal curves, giving a partial solution to the weak Schottky problem in this case. Second, we formulate questions from scattering amplitudes in a broader context using very affine varieties and D-module theory. The interplay between geometry and combinatorics in particle physics indeed suggests an underlying, coherent mathematical structure behind the study of particle interactions. In this thesis, we gain a better understanding of mathematical objects, such as moduli spaces of point configurations and generalized Euler integrals, for which particle physics provides concrete, non-trivial examples, and we prove some conjectures stated in the physics literature. Finally, we study linear spaces of symmetric matrices, addressing questions motivated by algebraic statistics, optimization, and enumerative geometry. This includes giving explicit formulas for the maximum likelihood degree and studying tangency problems for quadric surfaces in projective space from the point of view of real algebraic geometry

    Rational Function Simplification for Integration-by-Parts Reduction and Beyond

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    We present FUEL (Fractional Universal Evaluation Library), a C++ library for performing rational function arithmetic with a flexible choice of third-party computer algebra systems as simplifiers. FUEL is an outgrowth of a C++ interface to Fermat which was originally part of the FIRE code for integration-by-parts (IBP) reduction for Feynman integrals, now promoted to be a standalone library and with access to simplifiers other than Fermat. We compare the performance of various simplifiers for standalone benchmark problems as well as IBP reduction runs with FIRE.Comment: 18 pages, 1 figure, 6 table

    On a New, Efficient Framework for Falsifiable Non-interactive Zero-Knowledge Arguments

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    Et kunnskapslĂžst bevis er en protokoll mellom en bevisfĂžrer og en attestant. BevisfĂžreren har som mĂ„l Ă„ overbevise attestanten om at visse utsagn er korrekte, som besittelse av kortnummeret til et gyldig kredittkort, uten Ă„ avslĂžre noen private opplysninger, som for eksempel kortnummeret selv. I mange anvendelser er det Ăžnskelig Ă„ bruke IIK-bevis (Ikke-interaktive kunnskapslĂžse bevis), der bevisfĂžreren produserer kun en enkelt melding som kan bekreftes av mange attestanter. En ulempe er at sikre IIK-bevis for ikke-trivielle sprĂ„k kun kan eksistere ved tilstedevĂŠrelsen av en pĂ„litelig tredjepart som beregner en felles referansestreng som blir gjort tilgjengelig for bĂ„de bevisfĂžreren og attestanten. NĂ„r ingen slik part eksisterer liter man av og til pĂ„ ikke-interaktiv vitne-uskillbarhet, en svakere form for personvern. Studiet av effektive og sikre IIK-bevis er en kritisk del av kryptografi som har blomstret opp i det siste grunnet anvendelser i blokkjeder. I den fĂžrste artikkelen konstruerer vi et nytt IIK-bevis for sprĂ„kene som bestĂ„r av alle felles nullpunkter for en endelig mengde polynomer over en endelig kropp. Vi demonstrerer nytteverdien av beviset ved flerfoldige eksempler pĂ„ anvendelser. SĂŠrlig verdt Ă„ merke seg er at det er mulig Ă„ gĂ„ nesten automatisk fra en beskrivelse av et sprĂ„k pĂ„ et hĂžyt nivĂ„ til definisjonen av IIK-beviset, som minsker behovet for dedikert kryptografisk ekspertise. I den andre artikkelen konstruerer vi et IIV-bevis ved Ă„ bruke en ny kompilator. Vi utforsker begrepet Kunnskapslydighet (et sterkere sikkerhetsbegrep enn lydighet) for noen konstruksjoner av IIK-bevis. I den tredje artikkelen utvider vi arbeidet fra den fĂžrste artikkelen ved Ă„ konstruere et nytt IIK-bevis for mengde-medlemskap som lar oss bevise at et element ligger, eller ikke ligger, i den gitte mengden. Flere nye konstruksjoner har bedre effektivitet sammenlignet med allerede kjente konstruksjoner.A zero-knowledge proof is a protocol between a prover, and a verifier. The prover aims to convince the verifier of the truth of some statement, such as possessing credentials for a valid credit card, without revealing any private information, such as the credentials themselves. In many applications, it is desirable to use NIZKs (Non-Interactive Zero Knowledge) proofs, where the prover sends outputs only a single message that can be verified by many verifiers. As a drawback, secure NIZKs for non-trivial languages can only exist in the presence of a trusted third party that computes a common reference string and makes it available to both the prover and verifier. When no such party exists, one sometimes relies on non interactive witness indistinguishability (NIWI), a weaker notion of privacy. The study of efficient and secure NIZKs is a crucial part of cryptography that has been thriving recently due to blockchain applications. In the first paper, we construct a new NIZK for the language of common zeros of a finite set of polynomials over a finite field. We demonstrate its usefulness by giving a large number of example applications. Notably, it is possible to go from a high-level language description to the definition of the NIZK almost automatically, lessening the need for dedicated cryptographic expertise. In the second paper, we construct a NIWI using a new compiler. We explore the notion of Knowledge Soundness (a security notion stronger than soundness) of some NIZK constructions. In the third paper, we extended the first paper’s work by constructing a new set (non-)membership NIZK that allows us to prove that an element belongs or does not belong to the given set. Many new constructions have better efficiency compared to already-known constructions.Doktorgradsavhandlin

    LIPIcs, Volume 258, SoCG 2023, Complete Volume

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    LIPIcs, Volume 258, SoCG 2023, Complete Volum

    Environmental Effects of Stratospheric Ozone Depletion, UV Radiation, and interactions with Climate Change: 2022 Assessment Report

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    The Montreal Protocol on Substances that Deplete the Ozone Layer was established 35 years ago following the 1985 Vienna Convention for protection of the environment and human health against excessive amounts of harmful ultraviolet-B (UV-B, 280-315 nm) radiation reaching the Earth’s surface due to a reduced UV-B-absorbing ozone layer. The Montreal Protocol, ratified globally by all 198 Parties (countries), controls ca 100 ozone-depleting substances (ODS). These substances have been used in many applications, such as in refrigerants, air conditioners, aerosol propellants, fumigants against pests, fire extinguishers, and foam materials. The Montreal Protocol has phased out nearly 99% of ODS, including ODS with high global warming potentials such as chlorofluorocarbons (CFC), thus serving a dual purpose. However, some of the replacements for ODS also have high global warming potentials, for example, the hydrofluorocarbons (HFCs). Several of these replacements have been added to the substances controlled by the Montreal Protocol. The HFCs are now being phased down under the Kigali Amendment. As of December 2022, 145 countries have signed the Kigali Amendment, exemplifying key additional outcomes of the Montreal Protocol, namely, that of also curbing climate warming and stimulating innovations to increase energy efficiency of cooling equipment used industrially as well as domestically. As the concentrations of ODS decline in the upper atmosphere, the stratospheric ozone layer is projected to recover to pre-1980 levels by the middle of the 21st century, assuming full compliance with the control measures of the Montreal Protocol. However, in the coming decades, the ozone layer will be increasingly influenced by emissions of greenhouse gases and ensuing global warming. These trends are highly likely to modify the amount of UV radiation reaching the Earth\u27s surface with implications for the effects on ecosystems and human health. Against this background, four Panels of experts were established in 1988 to support and advise the Parties to the Montreal Protocol with up-to-date information to facilitate decisions for protecting the stratospheric ozone layer. In 1990 the four Panels were consolidated into three, the Scientific Assessment Panel, the Environmental Effects Assessment Panel, and the Technology and Economic Assessment Panel. Every four years, each of the Panels provides their Quadrennial Assessments as well as a Synthesis Report that summarises the key findings of all the Panels. In the in-between years leading up to the quadrennial, the Panels continue to inform the Parties to the Montreal Protocol of new scientific information

    Twisted cohomology and likelihood ideals

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    A likelihood function on a smooth very affine variety gives rise to a twisted de Rham complex. We show how its top cohomology vector space degenerates to the coordinate ring of the critical points defined by the likelihood equations. We obtain a basis for cohomology from a basis of this coordinate ring. We investigate the dual picture, where twisted cycles correspond to critical points. We show how to expand a twisted cocycle in terms of a basis, and apply our methods to Feynman integrals from physics.Comment: 28 pages, 2 figures, comments are welcom

    Algorithms in Intersection Theory in the Plane

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    This thesis presents an algorithm to find the local structure of intersections of plane curves. More precisely, we address the question of describing the scheme of the quotient ring of a bivariate zero-dimensional ideal I⊆K[x,y]I\subseteq \mathbb K[x,y], \textit{i.e.} finding the points (maximal ideals of K[x,y]/I\mathbb K[x,y]/I) and describing the regular functions on those points. A natural way to address this problem is via Gr\"obner bases as they reduce the problem of finding the points to a problem of factorisation, and the sheaf of rings of regular functions can be studied with those bases through the division algorithm and localisation. Let I⊆K[x,y]I\subseteq \mathbb K[x,y] be an ideal generated by F\mathcal F, a subset of A[x,y]\mathbb A[x,y] with Aâ†ȘK\mathbb A\hookrightarrow\mathbb K and K\mathbb K a field. We present an algorithm that features a quadratic convergence to find a Gr\"obner basis of II or its primary component at the origin. We introduce an m\mathfrak m-adic Newton iteration to lift the lexicographic Gr\"obner basis of any finite intersection of zero-dimensional primary components of II if m⊆A\mathfrak m\subseteq \mathbb A is a \textit{good} maximal ideal. It relies on a structural result about the syzygies in such a basis due to Conca \textit{\&} Valla (2008), from which arises an explicit map between ideals in a stratum (or Gr\"obner cell) and points in the associated moduli space. We also qualify what makes a maximal ideal m\mathfrak m suitable for our filtration. When the field K\mathbb K is \textit{large enough}, endowed with an Archimedean or ultrametric valuation, and admits a fraction reconstruction algorithm, we use this result to give a complete m\mathfrak m-adic algorithm to recover G\mathcal G, the Gr\"obner basis of II. We observe that previous results of Lazard that use Hermite normal forms to compute Gr\"obner bases for ideals with two generators can be generalised to a set of nn generators. We use this result to obtain a bound on the height of the coefficients of G\mathcal G and to control the probability of choosing a \textit{good} maximal ideal m⊆A\mathfrak m\subseteq\mathbb A to build the m\mathfrak m-adic expansion of G\mathcal G. Inspired by Pardue (1994), we also give a constructive proof to characterise a Zariski open set of GL2(K)\mathrm{GL}_2(\mathbb K) (with action on K[x,y]\mathbb K[x,y]) that changes coordinates in such a way as to ensure the initial term ideal of a zero-dimensional II becomes Borel-fixed when ∣K∣|\mathbb K| is sufficiently large. This sharpens our analysis to obtain, when A=Z\mathbb A=\mathbb Z or A=k[t]\mathbb A=k[t], a complexity less than cubic in terms of the dimension of Q[x,y]/⟹G⟩\mathbb Q[x,y]/\langle \mathcal G\rangle and softly linear in the height of the coefficients of G\mathcal G. We adapt the resulting method and present the analysis to find the ⟹x,y⟩\langle x,y\rangle-primary component of II. We also discuss the transition towards other primary components via linear mappings, called \emph{untangling} and \emph{tangling}, introduced by van der Hoeven and Lecerf (2017). The two maps form one isomorphism to find points with an isomorphic local structure and, at the origin, bind them. We give a slightly faster tangling algorithm and discuss new applications of these techniques. We show how to extend these ideas to bivariate settings and give a bound on the arithmetic complexity for certain algebras
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