Related papers: Expansivity theory and Sendov's conjecture

We study Polynomial Lawvere logic PL, a logic defined over the Lawvere quantale of extended positive reals with sum as tensor, to which we add multiplication, thereby obtaining a semiring structure. PL is designed for complex quantitative…

The polynomial Szemer\'{e}di theorem implies that, for any $\delta \in (0,1)$, any family $\{P_1,\ldots, P_m\} \subset \mathbb{Z}[y]$ of nonconstant polynomials with constant term zero, and any sufficiently large $N$, every subset of…

We classify all tuples of lattice polyhedra of relative mixed volume 1 and all minimal (by inclusion) tuples of polyhedra of relative mixed volume 2. We also prove a conjecture by A. Esterov, which states that all tuples with finite…

We define a sequence of positive integers recursively, where each term is determined as follows: starting with a given positive integer, if the term is odd, the next is the sum of its positive divisors; if the term is even, the subsequent…

Turing progressions have been often used to measure the proof-theoretic strength of mathematical theories. Turing progressions based on $n$-provability give rise to a $\Pi_{n+1}$ proof-theoretic ordinal. As such, to each theory $U$ we can…

We prove a theorem on distortion of cross ratio of four points under the mapping effected by a complex polynomial with restricted critical values. Its corollaries include inequalities involving the absolute value and certain coefficients of…

We restate the semistable reduction theorem from geometric invariant theory in the context of spaces of morphisms on $\mathbb{P}^{n}$. For every complete curve $C$ downstairs, we get a $\mathbb{P}^{n}$-bundle on an abstract curve $D$…

We generalise the expansion formulae of Musiker, Schiffler and Williams, obtained for cluster algebras from orientable surfaces, to a larger class of coefficients which we call principal laminations. In doing so, for any quasi-cluster…

We study the set of algebraic objects known as vanishing polynomials (the set of polynomials that annihilate all elements of a ring) over general commutative rings with identity. These objects are of special interest due to their close…

We estimate the number of zeros of a polynomial in $\mathbb{C}[z]$ within any small circular disc centered on the unit circle, which improves and comprehensively extends a result established by Borwein, Erd{\'e}lyi, and Littmann~\cite{BE1}…

According to the classical Gauss-Lucas theorem all zeros of the derivative of a complex non-constant polynomial p lie in the convex hull of the zeros of p. It is proved that for a polynomial p of degree four with four different zeros…

In the paper, the occurrence of zeros and ones in the binary expansion of the primes is studied. In particular the statement in the title is established. The proof is unconditional.

Determining the matrix multiplication exponent $\omega$ is one of the greatest open problems in theoretical computer science. We show that it is impossible to prove $\omega = 2$ by starting with structure tensors of modules of fixed degree…

Let ${\cal P}_n^c$ denote the set of all algebraic polynomials of degree at most $n$ with complex coefficients. Let $$D^+ := \{z \in \mathbb{C}: |z| \leq 1, \, \, \Im(z) \geq 0\}$$ be the closed upper half-disk of the complex plane. For…

Let $P_1,\dots,P_m\in\mathbb{Z}[y]$ be any linearly independent polynomials with zero constant term. We show that there exists a $\gamma>0$ such that any subset of $\mathbb{F}_q$ of size at least $q^{1-\gamma}$ contains a nontrivial…

The separability tensor element of a separable extension of noncommutative rings is an idempotent when viewed in the correct endomorphism ring; so one speaks of a separability idempotent, as one usually does for separable algebras. It is…

We generalize the polynomial Szemer\'{e}di theorem to intersective polynomials over the ring of integers of an algebraic number field, by which we mean polynomials having a common root modulo every ideal. This leads to the existence of new…

We consider polynomials in R[x] which map the set of nonnegative (element-wise) matrices of a given order into itself. Let n be a positive integer and define P(n)= {p in R[x] : p(A) is nonnegative (element-wise), for all A, A an n-by-n…

We derive sharp Sobolev embeddings on a class of Sobolev spaces with potential weights without assuming any boundary conditions. Moreover, we consider the Adams-type inequalities for the borderline Sobolev embedding into the exponential…

We introduce a collection of polynomials $F_N$, associated to each positive integer $N$, whose divisibility properties yield a reformulation of the Goldbach conjecture. While this reformulation certainly does not lead to a resolution of the…