On Necessary and Sufficient Conditions for Differential Flatness

This paper is devoted to the characterization of differentially flat nonlinear systems in implicit representation, after elimination of the input variables, in the differential geometric framework of manifolds of jets of infinite order. We extend the notion of Lie-B\"acklund equivalence, introduced in Fliess et al. (1999), to this implicit context and focus attention on Lie-B\"acklund isomorphisms associated to flat systems, called trivializations. They can be locally characterized in terms of polynomial matrices of the indeterminate \ddt, whose range is equal to the kernel of the polynomial matrix associated to the implicit variational system. Such polynomial matrices are useful to compute the ideal of differential forms generated by the differentials of all possible trivializations. We introduce the notion of a strongly closed ideal of differential forms, and prove that flatness is equivalent to the strong closedness of the latter ideal, which, in turn, is equivalent to the existence of solutions of the so-called generalized moving frame structure equations. Two sequential procedures to effectively compute flat outputs are deduced and various examples and consequences are presented.Comment: Version 3 is the published versio

Semi-regular sequences and other random systems of equations

The security of multivariate cryptosystems and digital signature schemes relies on the hardness of solving a system of polynomial equations over a finite field. Polynomial system solving is also currently a bottleneck of index-calculus algorithms to solve the elliptic and hyperelliptic curve discrete logarithm problem. The complexity of solving a system of polynomial equations is closely related to the cost of computing Groebner bases, since computing the solutions of a polynomial system can be reduced to finding a lexicographic Groebner basis for the ideal generated by the equations. Several algorithms for computing such bases exist: We consider those based on repeated Gaussian elimination of Macaulay matrices. In this paper, we analyze the case of random systems, where random systems means either semi-regular systems, or quadratic systems in n variables which contain a regular sequence of n polynomials. We provide explicit formulae for bounds on the solving degree of semi-regular systems with m > n equations in n variables, for equations of arbitrary degrees for m = n+1, and for any m for systems of quadratic or cubic polynomials. In the appendix, we provide a table of bounds for the solving degree of semi-regular systems of m = n + k quadratic equations in n variables for 2 <= k; n <= 100 and online we provide the values of the bounds for 2 <= k; n <= 500. For quadratic systems which contain a regular sequence of n polynomials, we argue that the Eisenbud-Green-Harris Conjecture, if true, provides a sharp bound for their solving degree, which we compute explicitly.Comment: 27 pages, 4 table

Variable elimination in chemical reaction networks with mass action kinetics

We consider chemical reaction networks taken with mass action kinetics. The steady states of such a system are solutions to a system of polynomial equations. Even for small systems the task of finding the solutions is daunting. We develop an algebraic framework and procedure for linear elimination of variables. The procedure reduces the variables in the system to a set of "core" variables by eliminating variables corresponding to a set of non-interacting species. The steady states are parameterized algebraically by the core variables, and a graphical condition is given for when a steady state with positive core variables necessarily have all variables positive. Further, we characterize graphically the sets of eliminated variables that are constrained by a conservation law and show that this conservation law takes a specific form

A clever elimination strategy for efficient minimal solvers

We present a new insight into the systematic generation of minimal solvers in computer vision, which leads to smaller and faster solvers. Many minimal problem formulations are coupled sets of linear and polynomial equations where image measurements enter the linear equations only. We show that it is useful to solve such systems by first eliminating all the unknowns that do not appear in the linear equations and then extending solutions to the rest of unknowns. This can be generalized to fully non-linear systems by linearization via lifting. We demonstrate that this approach leads to more efficient solvers in three problems of partially calibrated relative camera pose computation with unknown focal length and/or radial distortion. Our approach also generates new interesting constraints on the fundamental matrices of partially calibrated cameras, which were not known before.Comment: 13 pages, 7 figure

The Structure of Differential Invariants and Differential Cut Elimination

The biggest challenge in hybrid systems verification is the handling of differential equations. Because computable closed-form solutions only exist for very simple differential equations, proof certificates have been proposed for more scalable verification. Search procedures for these proof certificates are still rather ad-hoc, though, because the problem structure is only understood poorly. We investigate differential invariants, which define an induction principle for differential equations and which can be checked for invariance along a differential equation just by using their differential structure, without having to solve them. We study the structural properties of differential invariants. To analyze trade-offs for proof search complexity, we identify more than a dozen relations between several classes of differential invariants and compare their deductive power. As our main results, we analyze the deductive power of differential cuts and the deductive power of differential invariants with auxiliary differential variables. We refute the differential cut elimination hypothesis and show that, unlike standard cuts, differential cuts are fundamental proof principles that strictly increase the deductive power. We also prove that the deductive power increases further when adding auxiliary differential variables to the dynamics

Eliminating Variables in Boolean Equation Systems

Systems of Boolean equations of low degree arise in a natural way when analyzing block ciphers. The cipher's round functions relate the secret key to auxiliary variables that are introduced by each successive round. In algebraic cryptanalysis, the attacker attempts to solve the resulting equation system in order to extract the secret key. In this paper we study algorithms for eliminating the auxiliary variables from these systems of Boolean equations. It is known that elimination of variables in general increases the degree of the equations involved. In order to contain computational complexity and storage complexity, we present two new algorithms for performing elimination while bounding the degree at $3$, which is the lowest possible for elimination. Further we show that the new algorithms are related to the well known \emph{XL} algorithm. We apply the algorithms to a downscaled version of the LowMC cipher and to a toy cipher based on the Prince cipher, and report on experimental results pertaining to these examples.Comment: 21 pages, 3 figures, Journal pape

Exploiting chordal structure in polynomial ideals: a Gr\"obner bases approach

Chordal structure and bounded treewidth allow for efficient computation in numerical linear algebra, graphical models, constraint satisfaction and many other areas. In this paper, we begin the study of how to exploit chordal structure in computational algebraic geometry, and in particular, for solving polynomial systems. The structure of a system of polynomial equations can be described in terms of a graph. By carefully exploiting the properties of this graph (in particular, its chordal completions), more efficient algorithms can be developed. To this end, we develop a new technique, which we refer to as chordal elimination, that relies on elimination theory and Gr\"obner bases. By maintaining graph structure throughout the process, chordal elimination can outperform standard Gr\"obner basis algorithms in many cases. The reason is that all computations are done on "smaller" rings, of size equal to the treewidth of the graph. In particular, for a restricted class of ideals, the computational complexity is linear in the number of variables. Chordal structure arises in many relevant applications. We demonstrate the suitability of our methods in examples from graph colorings, cryptography, sensor localization and differential equations.Comment: 40 pages, 5 figure