Related papers: A Generalization of the B\^{o}cher-Grace Theorem

We present a method to compute the Euler characteristic of an algebraic subset of $\bc^n$. This method relies on clasical tools such as Gr\"obner basis and primary decomposition. The existence of this method allows us to define a new…

We present a new proof of the necessary and sufficient condition for the existence of a triangle that is simultaneously inscribed in a circle and circumscribed about a central conic (an ellipse or a hyperbola). In the limiting case where…

The critical point degree of a periodic graph operator is the number of critical points of its complex Bloch variety. Determining it is a step towards the spectral edges conjecture and more generally understanding Bloch varieties. Previous…

The Upper Bound Theorem for convex polytopes implies that the $p$-th Betti number of the \v{C}ech complex of any set of $N$ points in $\mathbb R^d$ and any radius satisfies $\beta_{p} = O(N^{m})$, with $m = \min \{ p+1, \lceil d/2 \rceil…

Using the Moore--Penrose pseudoinverse, this work generalizes the gradient approximation technique called centred simplex gradient to allow sample sets containing any number of points. This approximation technique is called the…

For univariate polynomials over arbitrary field the degree gives an upper bound on the number of roots (factor theorem) and as a related result for any finite point-set one can construct a polynomial of degree equal to the cardinality…

In this article we prove a bifurcation and multiplicity result for a critical problem involving a degenerate nonlinear operator $\Delta_\gamma^p$. We extend to a generic $p>1$ a result which was proved only when $p=2$. When $p\neq 2$, the…

A generalization of the term "generalized Clifford algebras" (as appears in papers on advances in applied Clifford algebras) is introduced. This algebra is studied by means of structure theory of central simple algebras. A graph theoretical…

We present higher dimensional versions of the classical results of Euler and Fuss, both of which are special cases of the celebrated Poncelet porism. Our results concern polytopes, specifically simplices, parallelotopes and cross polytopes,…

The degree partition of a simple graph is its degree sequence rearranged in weakly decreasing order. The polytope of degree partitions (respectively, degree sequences) is the convex hull of all degree partitions (respectively, degree…

Let f be a real or complex polynomial. We give an algorithm to compute the set of generalized critical values. The algorithm uses a finite dimensional space of rational arcs along which we can reach all generalized critical values of f.

Section 10.4 of the 1998 Springer-Verlag book {\em Complexity and Real Computation}, by Blum, Cucker, Shub, and Smale, contains a particularly elegant proof of the Fundamental Theorem of Algebra: The central idea of the proof naturally…

Let V $\subset$ C n be an equidimensional algebraic set and g be an n-variate polynomial with rational coefficients. Computing the critical points of the map that evaluates g at the points of V is a cornerstone of several algorithms in real…

Let $p$ be a prime. Let $A$ and $B$, $A \ge B \ge 0$, be integers with base $p$ expansions $A = \alpha_i\alpha_{i-1}\dots \alpha_0$ and $B = \beta_i\beta_{i-1}\dots \beta_0$. Lucas proved that $$\binom{A}{B} \equiv…

We propose 3D generalizations of the Feuerbach theorem: the first one deals with a tetrahedron analogue of the Euler circle, the second one is done by means of an {\guillemotleft}up-in-ex-touch{\guillemotright} construction. Then we give a…

The centerpoint theorem is a well-known and widely used result in discrete geometry. It states that for any point set $P$ of $n$ points in $\mathbb{R}^d$, there is a point $c$, not necessarily from $P$, such that each halfspace containing…

Let $P$ be a polytope with rational vertices. A classical theorem of Ehrhart states that the number of lattice points in the dilations $P(n) = nP$ is a quasi-polynomial in $n$. We generalize this theorem by allowing the vertices of P(n) to…

This paper studies a class of binomial ideals associated to graphs with finite vertex sets. They generalize the binomial edge ideals, and they arise in the study of conditional independence ideals. A Gr\"obner basis can be computed by…

We consider the convex set $\Gamma_{m,n}$ of $m\times n$ stochastic matrices and the convex set $\Gamma_{m,n}^\pi\subset \Gamma_{m,n}$ of $m\times n$ centrosymmetric stochastic matrices (stochastic matrices that are symmetric under rotation…

Carath\'eodory showed that $n$ complex numbers $c_1,...,c_n$ can uniquely be written in the form $c_p=\sum_{j=1}^m \rho_j {\epsilon_j}^p$ with $p=1,...,n$, where the $\epsilon_j$s are different unimodular complex numbers, the $\rho_j$s are…