Local Bézout Theorem

AbstractWe give an elementary proof of what we call the Local Bézout Theorem. Given a system of n polynomials in n indeterminates with coefficients in a Henselian local domain, (V,m,k), which residually defines an isolated point in kn of multiplicity r, we prove (under some additional hypothesis on V) that there are finitely many zeroes of the system above the residual zero (i.e., with coordinates in m), and the sum of their multiplicities is r. Our proof is based on techniques of computational algebra

Fast integer multiplication using generalized Fermat primes

For almost 35 years, Sch{\"o}nhage-Strassen's algorithm has been the fastest algorithm known for multiplying integers, with a time complexity O(n $\times$ log n $\times$ log log n) for multiplying n-bit inputs. In 2007, F{\"u}rer proved that there exists K > 1 and an algorithm performing this operation in O(n $\times$ log n $\times$ K log n). Recent work by Harvey, van der Hoeven, and Lecerf showed that this complexity estimate can be improved in order to get K = 8, and conjecturally K = 4. Using an alternative algorithm, which relies on arithmetic modulo generalized Fermat primes, we obtain conjecturally the same result K = 4 via a careful complexity analysis in the deterministic multitape Turing model

Conditional and approximate symmetries for nonlinear partial differential equations

M.Sc.In this work we concentrate on two generalizations of Lie symmetries namely conditional symmetries in the form of Q-symmetries and approximate symmetries. The theorems and definitions presented can be used to obtain exact and approximate solutions for nonlinear partial differential equations. These are then applied to various nonlinear heat and wave equations and many interesting solutions are given. Chapters 1 and 2 gives an introduction to the classical Lie approach. Chapters 3, 4 and 5 deals with conditional -, approximate -, and approximate conditional symmetries respectively. In chapter 6 we give a review of symbolic algebra computer packages available to aid in the search for symmetries, as well as useful REDUCE programs which were written to obtain the results given in chapters 2 to 5

Fast norm computation in smooth-degree Abelian number fields

This paper presents a fast method to compute algebraic norms of integral elements of smooth-degree cyclotomic fields, and, more generally, smooth-degree Galois number fields with commutative Galois groups. The typical scenario arising in $S$-unit searches (for, e.g., class-group computation) is computing a $\Theta(n\log n)$-bit norm of an element of weight $n^{1/2+o(1)}$ in a degree-$n$ field; this method then uses $n(\log n)^{3+o(1)}$ bit operations. An $n(\log n)^{O(1)}$ operation count was already known in two easier special cases: norms from power-of-2 cyclotomic fields via towers of power-of-2 cyclotomic subfields, and norms from multiquadratic fields via towers of multiquadratic subfields. This paper handles more general Abelian fields by identifying tower-compatible integral bases supporting fast multiplication; in particular, there is a synergy between tower-compatible Gauss-period integral bases and a fast-multiplication idea from Rader. As a baseline, this paper also analyzes various standard norm-computation techniques that apply to arbitrary number fields, concluding that all of these techniques use at least $n^2(\log n)^{2+o(1)}$ bit operations in the same scenario, even with fast subroutines for continued fractions and for complex FFTs. Compared to this baseline, algorithms dedicated to smooth-degree Abelian fields find each norm $n/(\log n)^{1+o(1)}$ times faster, and finish norm computations inside $S$-unit searches $n^2/(\log n)^{1+o(1)}$ times faster

Efficient NIZKs for Algebraic Sets

Significantly extending the framework of (Couteau and Hartmann, Crypto 2020), we propose a general methodology to construct NIZKs for showing that an encrypted vector $\vec{\chi}$ belongs to an algebraic set, i.e., is in the zero locus of an ideal $\mathscr{I}$ of a polynomial ring. In the case where $\mathscr{I}$ is principal, i.e., generated by a single polynomial $F$, we first construct a matrix that is a ``quasideterminantal representation\u27\u27 of $F$ and then a NIZK argument to show that $F (\vec{\chi}) = 0$. This leads to compact NIZKs for general computational structures, such as polynomial-size algebraic branching programs. We extend the framework to the case where \IDEAL is non-principal, obtaining efficient NIZKs for R1CS, arithmetic constraint satisfaction systems, and thus for $\mathsf{NP}$. As an independent result, we explicitly describe the corresponding language of ciphertexts as an algebraic language, with smaller parameters than in previous constructions that were based on the disjunction of algebraic languages. This results in an efficient GL-SPHF for algebraic branching programs

Alternative pharmaceuticals: The technoscientific becomings of Tibetan medicines in-between India and Switzerland

This doctoral dissertation forges and explores connections, flows and frictions between two seemingly unrelated manufacturers of Tibetan medicines: Men-Tsee-Khang, the Tibetan Medical and Astrological Institute in Dharamsala (Himachal Pradesh, India), and PADMA AG in Wetzikon (Zürich, Switzerland). Adopting a translocal, multispecies approach by positioning plant-medicines as the central actors in this ethnography, I trace how four plants - aru, ruta, tserngön and bongnak - become part of medicine in and between these two establishments of Sowa Rigpa of similar age and output volume, situated in highly diverse contexts at a stereotypical 'periphery' and 'core' of Western technoscience respectively. Inspired by Science and Technology Studies and by Pordié and Gaudillière's (2014a) 'reformulation regime' of industrial Ayurvedic proprietary products, I analyse the on-going material, technoscientific, and regulatory reformulations of Tibetan materia medica as they are actualised in contemporary recipes based on classical texts. In this thesis, I describe how both PADMA and Men-Tsee-Khang refer to Tibetan medical texts yet also rely on botanical taxonomy for plant identification. Both face the uncertainties of sourcing raw materials in bulk from growers and traders on the Indian market, skilfully mass-produce pills by means of machines for grinding, mixing, sieving and packaging, and depend on in-house laboratory analyses and each-other's expertise in the construction of hybrid 'qualities'. They are also forced to interact with technomedical conceptions of drug safety and toxicity, and with European medicine and food registration legislation to varying degrees. I argue that in performing this series of technoscientific reformulations, Tibetan medicines are becoming 'alternative pharmaceuticals': liminal, paradoxical yet politically subversive things oscillating betwixt and between tradition and modernity, orthodoxy and innovation, East and West. Men-Tsee-Khang and PADMA could thus be interpreted as two possible instantiations of a quasi-industrial techno-Sowa Rigpa, but only if one distinguishes 'Big' from 'Small Alternative' Pharma, and never without leaving crucial contradictions and identity politics behind