Related papers: Lecture on the combinatorial algebraic method for …
In this paper, we propose a new algebraic winding number and prove that it computes the number of complex roots of a polynomial in a rectangle, including roots on edges or vertices with appropriate counting. The definition makes sense for…
In this paper we give several conditions implying the irreducibility of the algebraic curve P(x)-Q(y)=0, where P,Q are rational functions. We also apply the results obtained to the functional equations P(f)=Q(g) and P(f)=cP(g), where c\in…
The Quantum Inverse Scattering Method is a scheme for solving integrable models in $1+1$ dimensions, building on an $R$-matrix that satisfies the Yang--Baxter equation and in terms of which one constructs a commuting family of transfer…
The goal of this paper is to measure the non-convexity of compact and smooth connected components of real algebraic plane curves. We study these curves first in a general setting and then in an asymptotic one. In particular, we consider…
P-algebras are a non-commutative, non-associative generalization of Boolean algebras that are for quantum logic what Boolean algebras are for classical logic. P-algebras have type <X, 0, ', .> where 0 is a constant, ' is unary and . is…
Using elementary linear algebra, this paper clarifies and proves some concepts about a recently introduced octonion-like associative division algebra over R. This octonion-like algebra is actually the same as the split-biquaternion algebra,…
Two sets of conditions are presented for the compactness of a real plane algebraic curve, one sufficient and one necessary, in terms of the Newton polygon of the defining polynomial.
The computation of the topology of a real algebraic plane curve is greatly simplified if there are no more than one critical point in each vertical line: the general position condition. When this condition is not satisfied, then a finite…
In this thesis quadratic and cubic algebras, which are extensions of SU(1,1) and SU(2) are studied in detail, with particular attention being given to their construction, their finite and infinite dimensional irreducible representations and…
In quantum field theory the path integral is usually formulated in the wave picture, i.e., as a sum over field evolutions. This path integral is difficult to define rigorously because of analytic problems whose resolution may ultimately…
Let $G$ be a complex classical group, and let $V$ be its defining representation (possibly plus a copy of the dual). A foundational problem in classical invariant theory is to write down generators and relations for the ring of…
A classical problem in Distance Geometry, with multiple practical applications (in molecular structure determination, sensor network localization etc.) is to find the possible placements of the vertices of a graph with given edge lengths.…
A non-zero constant Jacobian polynomial maps $F=(P,Q)$ of $\mathbb{C}^2$ is invertible if $P$ and $Q$ are rational polynomials.
Computing the determinant of a matrix with the univariate and multivariate polynomial entries arises frequently in the scientific computing and engineering fields. In this paper, an effective algorithm is presented for computing the…
In this article we give an implementation of the standard algorithm to segment a real algebraic plane curve defined implicitly. Our implementation is efficient and simpler than previous. We use global information to count the number of…
Proofs of the fundamental theorem of algebra can be divided up into three groups according to the techniques involved: proofs that rely on real or complex analysis, algebraic proofs, and topological proofs. Algebraic proofs make use of the…
We present the Polar framework for fully automating the analysis of classical and probabilistic loops using algebraic reasoning. The central theme in Polar comes with handling algebraic recurrences that precisely capture the loop semantics.…
A purely numerical approach to compact Riemann surfaces starting from plane algebraic curves is presented. The critical points of the algebraic curve are computed via a two-dimensional Newton iteration. The starting values for this…
We give combinatorial formulas for F-polynomials in cluster algebras of classical types in terms of the weighted paths in certain directed graphs. As a consequence we prove the positivity of F-polynomials in cluster algebras of classical…
We enumerate plane complex algebraic curves of a given degree with one singularity of any given topological type. Our approach is to compute the homology classes of the corresponding equisingular strata in the parameter spaces of plane…