A counterexample to a conjecture of Bass, Connell and Wright

Piotr Ossowski

Displaying similar documents to “A counterexample to a conjecture of Bass, Connell and Wright”

On the generalized vanishing conjecture

Zhenzhen Feng, Xiaosong Sun (2019)

Czechoslovak Mathematical Journal

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We show that the GVC (generalized vanishing conjecture) holds for the differential operator $Λ = (\partial_{x} - Φ (\partial_{y})) \partial_{y}$ and all polynomials $P (x, y)$ , where $Φ (t)$ is any polynomial over the base field. The GVC arose from the study of the Jacobian conjecture.

The Jacobian Conjecture in case of "non-negative coefficients"

Ludwik M. Drużkowski (1997)

Annales Polonici Mathematici

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It is known that it is sufficient to consider in the Jacobian Conjecture only polynomial mappings of the form $F (x ₁, . . ., x_{n}) = x - H (x) : = (x ₁ - H ₁ (x ₁, . . ., x_{n}), . . ., x_{n} - H_{n} (x ₁, . . ., x_{n}))$ , where $H_{j}$ are homogeneous polynomials of degree 3 with real coefficients (or $H_{j} = 0$ ), j = 1,...,n and H’(x) is a nilpotent matrix for each $x = (x ₁, . . ., x_{n}) \in ℝ^{n}$ . We give another proof of Yu’s theorem that in the case of non-negative coefficients of H the mapping F is a polynomial automorphism, and we moreover prove that in that case $d e g F^{- 1} \leq {(d e g F)}^{i n d F - 1}$ , where $i n d F : = m a x i n d H^{'} (x) : x \in ℝ^{n}$ . Note that the above inequality is not true when the coefficients...

A note on the cubical dimension of new classes of binary trees

Kamal Kabyl, Abdelhafid Berrachedi, Éric Sopena (2015)

Czechoslovak Mathematical Journal

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The cubical dimension of a graph $G$ is the smallest dimension of a hypercube into which $G$ is embeddable as a subgraph. The conjecture of Havel (1984) claims that the cubical dimension of every balanced binary tree with $2^{n}$ vertices, $n \geq 1$ , is $n$ . The 2-rooted complete binary tree of depth $n$ is obtained from two copies of the complete binary tree of depth $n$ by adding an edge linking their respective roots. In this paper, we determine the cubical dimension of trees obtained by subdividing twice...

Recent progress on the Jacobian Conjecture

Michiel de Bondt, Arno van den Essen (2005)

Annales Polonici Mathematici

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We describe some recent developments concerning the Jacobian Conjecture (JC). First we describe Drużkowski’s result in [6] which asserts that it suffices to study the JC for Drużkowski mappings of the form $x + {(A x)}^{* 3}$ with A² = 0. Then we describe the authors’ result of [2] which asserts that it suffices to study the JC for so-called gradient mappings, i.e. mappings of the form x - ∇f, with $f \in k^{[n]}$ homogeneous of degree 4. Using this result we explain Zhao’s reformulation of the JC which asserts the...

A geometric construction for spectrally arbitrary sign pattern matrices and the $2 n$ -conjecture

Dipak Jadhav, Rajendra Deore (2023)

Czechoslovak Mathematical Journal

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We develop a geometric method for studying the spectral arbitrariness of a given sign pattern matrix. The method also provides a computational way of computing matrix realizations for a given characteristic polynomial. We also provide a partial answer to $2 n$ -conjecture. We determine that the $2 n$ -conjecture holds for the class of spectrally arbitrary patterns that have a column or row with at least $n - 1$ nonzero entries.

Integer points on a curve and the plane Jacobian problem

Nguyen Van Chau (2006)

Annales Polonici Mathematici

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A polynomial map F = (P,Q) ∈ ℤ[x,y]² with Jacobian $J F : = P_{x} Q_{y} - P_{y} Q_{x} \equiv 1$ has a polynomial inverse with integer coefficients if the complex plane curve P = 0 has infinitely many integer points.

Positively homogeneous functions and the Łojasiewicz gradient inequality

Alain Haraux (2005)

Annales Polonici Mathematici

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It is quite natural to conjecture that a positively homogeneous function with degree d ≥ 2 on $ℝ^{N}$ satisfies the Łojasiewicz gradient inequality with exponent θ = 1/d without any need for an analyticity assumption. We show that this property is true under some additional hypotheses, but not always, even for N = 2.

Order of the smallest counterexample to Gallai's conjecture

Fuyuan Chen (2018)

Czechoslovak Mathematical Journal

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In 1966, Gallai conjectured that all the longest paths of a connected graph have a common vertex. Zamfirescu conjectured that the smallest counterexample to Gallai’s conjecture is a graph on 12 vertices. We prove that Gallai’s conjecture is true for every connected graph $G$ with $α^{'} (G) \leq 5$ , which implies that Zamfirescu’s conjecture is true.

The strength of the projective Martin conjecture

C. T. Chong, Wei Wang, Liang Yu (2010)

Fundamenta Mathematicae

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We show that Martin’s conjecture on Π¹₁ functions uniformly $\leq_{T}$ -order preserving on a cone implies Π¹₁ Turing Determinacy over ZF + DC. In addition, it is also proved that for n ≥ 0, this conjecture for uniformly degree invariant $Π ¹_{2 n + 1}$ functions is equivalent over ZFC to $Σ ¹_{2 n + 2}$ -Axiom of Determinacy. As a corollary, the consistency of the conjecture for uniformly degree invariant Π¹₁ functions implies the consistency of the existence of a Woodin cardinal.

A proof of the Grünbaum conjecture

Bruce L. Chalmers, Grzegorz Lewicki (2010)

Studia Mathematica

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Let V be an n-dimensional real Banach space and let λ(V) denote its absolute projection constant. For any N ∈ N with N ≥ n define $λ ₙ^{N} = s u p λ (V) : d i m (V) = n, V \subset l_{\infty}^{(N)}$ , λₙ = supλ(V): dim(V) = n. A well-known Grünbaum conjecture [Trans. Amer. Math. Soc. 95 (1960)] says that λ₂ = 4/3. König and Tomczak-Jaegermann [J. Funct. Anal. 119 (1994)] made an attempt to prove this conjecture. Unfortunately, their Proposition 3.1, used in the proof, is incorrect. In this paper a complete proof of the Grünbaum conjecture is presented ...

On a number theoretic conjecture on positive integral points in a 5-dimensional tetrahedron and a sharp estimate of the Dickman–De Bruijn function

Ke-Pao Lin, Xue Luo, Stephen S.-T. Yau, Huaiqing Zuo (2014)

Journal of the European Mathematical Society

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It is well known that getting the estimate of integral points in right-angled simplices is equivalent to getting the estimate of Dickman-De Bruijn function $ψ (x, y)$ which is the number of positive integers $\leq x$ and free of prime factors $> y$ . Motivating from the Yau Geometry Conjecture, the third author formulated the Number Theoretic Conjecture which gives a sharp polynomial upper estimate that counts the number of positive integral points in n-dimensional ( $n \geq 3$ ) real right-angled simplices. In this...

On the Brocard-Ramanujan problem and generalizations

Andrzej Dąbrowski (2012)

Colloquium Mathematicae

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Let $p_{i}$ denote the ith prime. We conjecture that there are precisely 28 solutions to the equation $n ² - 1 = p ₁^{α ₁} \dots p_{k}^{α_{k}}$ in positive integers n and α₁,..., $α_{k}$ . This conjecture implies an explicit description of the set of solutions to the Brocard-Ramanujan equation. We also propose another variant of the Brocard-Ramanujan problem: describe the set of solutions in non-negative integers of the equation n! + A = x₁²+x₂²+x₃² (A fixed).

Dynamically defined Cantor sets under the conditions of McDuff's conjecture

Jorge Iglesias, Aldo Portela (2010)

Colloquium Mathematicae

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We prove that if the Cantor set K, dynamically defined by a function $S \in C^{1 + α}$ , satisfies the conditions of McDuff’s conjecture then it cannot be C¹-minimal.

Greenberg’s conjecture for $ℤ_{p}^{d}$ -extensions

Andrea Bandini (2003)

Acta Arithmetica

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Results related to Huppert’s $ρ$ - $σ$ conjecture

Xia Xu, Yong Yang (2023)

Czechoslovak Mathematical Journal

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We improve a few results related to Huppert’s $ρ$ - $σ$ conjecture. We also generalize a result about the covering number of character degrees to arbitrary finite groups.

Semifields and a theorem of Abhyankar

Vítězslav Kala (2017)

Commentationes Mathematicae Universitatis Carolinae

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Abhyankar proved that every field of finite transcendence degree over $ℚ$ or over a finite field is a homomorphic image of a subring of the ring of polynomials $ℤ [T_{1}, \dots, T_{n}]$ (for some $n$ depending on the field). We conjecture that his result cannot be substantially strengthened and show that our conjecture implies a well-known conjecture on the additive idempotence of semifields that are finitely generated as semirings.