Tootfinder

Tropical Fermat-Weber Polytopes
David Barnhill, John Sabol, Ruriko Yoshida, Keiji Miura
https://arxiv.org/abs/2402.14287 https://arxi…

Tropical Fermat-Weber Polytopes
We study the geometry of tropical Fermat-Weber points in terms of the symmetric tropical metric over the tropical projective torus. It is well known that a tropical Fermat-Weber point of a given sample is not unique and in this paper we show that the set of all possible Fermat-Weber points forms a polytrope. Then, we introduce the tropical Fermat-Weber gradient and using them, we show that the tropical Fermat-Weber polytrope is a bounded cell of a tropical hyperplane arrangement given by both m…

Asymptotic Fermat's Last Theorem for a family of equations of signature $(2, 2n, n)$
Pedro-Jos\'e Cazorla Garc\'ia
https://arxiv.org/abs/2404.14098

Asymptotic Fermat's Last Theorem for a family of equations of signature $(2, 2n, n)$
In this paper, we study the integer solutions of a family of Fermat-type equations of signature $(2, 2n, n)$, $Cx^2 + q^ky^{2n} = z^n$. We provide an algorithmically testable set of conditions which, if satisfied, imply the existence of a constant $B_{C, q}$ such that if $n > B_{C,q}$, there are no solutions $(x, y, z, n)$ of the equation. Our methods use the modular method for Diophantine equations, along with level lowering and Galois theory.

Eisenstein integers and equilateral ideal triangles
Greg McShane
https://arxiv.org/abs/2403.14375 https://arxiv.org/pdf/2403.14375
arXiv:2403.14375v1 Announce Type: new
Abstract: We discuss the relationship between Penner's $\lambda$-length and the norms of Eisenstein integers. This leads to a geometric proof of the fact, attributed to Fermat, that every prime $p$ of the form $3k 1$ is the norm of an Eisenstein integer that is can be written as $a^2 - ab b^2$ for some $a,b \in \mathbb{Z}$.

A Pythagorean triangle in which the hypotenuse and the sum of the arms are squares
Djamel Himane
https://arxiv.org/abs/2404.12906 https://

A Pythagorean triangle in which the hypotenuse and the sum of the arms are squares
In this paper, show that the Diophantine equation $ x^2+(x+1)^2=w^4 $ has only two solutions $ (0,1) $ and $ (119,13)$ in non-negative integers $ x $ and $ w $. This equation concerned a classic problem posed by Pierre de Fermat, wonders about finding a Pythagorean triangle in which the hypotenuse and the sum of the arms are square. We review the method of finding the smallest solution presented by Fermat, and the relationship between the primitive Pythagorean triples and the Pell's equation, F…

This https://arxiv.org/abs/2403.13311 has been replaced.
link: https://scholar.google.com/scholar?q=a

Multi-Robot Connected Fermat Spiral Coverage
We introduce the Multi-Robot Connected Fermat Spiral (MCFS), a novel algorithmic framework for Multi-Robot Coverage Path Planning (MCPP) that adapts Connected Fermat Spiral (CFS) from the computer graphics community to multi-robot coordination for the first time. MCFS uniquely enables the orchestration of multiple robots to generate coverage paths that contour around arbitrarily shaped obstacles, a feature that is notably lacking in traditional methods. Our framework not only enhances area cove…

All meromorphic solutions of Fermat-type functional equations
Feng L\"u
https://arxiv.org/abs/2404.10248 https://arxiv.org/pdf/2…

All meromorphic solutions of Fermat-type functional equations
In this paper, by making use of properties of elliptic functions, we describe meromorphic solutions of Fermat-type functional equations $f(z)^{n}+f(L(z))^{m}=1$ over the complex plane $\mathbb{C}$, where $L(z)$ is a nonconstant entire function, $m$ and $n$ are two positive integers. As applications, we also consider meromorphic solutions of Fermat-type difference and $q$-difference equations.

A Pythagorean triangle in which the hypotenuse and the sum of the arms are squares
Djamel Himane
https://arxiv.org/abs/2404.12906 https://

A Pythagorean triangle in which the hypotenuse and the sum of the arms are squares
In this paper, show that the Diophantine equation $ x^2+(x+1)^2=w^4 $ has only two solutions $ (0,1) $ and $ (119,13)$ in non-negative integers $ x $ and $ w $. This equation concerned a classic problem posed by Pierre de Fermat, wonders about finding a Pythagorean triangle in which the hypotenuse and the sum of the arms are square. We review the method of finding the smallest solution presented by Fermat, and the relationship between the primitive Pythagorean triples and the Pell's equation, F…

Combining Statistical Depth and Fermat Distance for Uncertainty Quantification
Hai-Vy Nguyen, Fabrice Gamboa, Reda Chhaibi, Sixin Zhang, Serge Gratton, Thierry Giaccone
https://arxiv.org/abs/2404.08476

Combining Statistical Depth and Fermat Distance for Uncertainty Quantification
We measure the Out-of-domain uncertainty in the prediction of Neural Networks using a statistical notion called ``Lens Depth'' (LD) combined with Fermat Distance, which is able to capture precisely the ``depth'' of a point with respect to a distribution in feature space, without any assumption about the form of distribution. Our method has no trainable parameter. The method is applicable to any classification model as it is applied directly in feature space at test time and does not intervene i…

On the solutions of the generalized Fermat equation over totally real number fields
Satyabrat Sahoo
https://arxiv.org/abs/2404.09171 https://

On the solutions of the generalized Fermat equation over totally real number fields
Let $K$ be a totally real number field, and $ \mathcal{O}_K$ be the ring of integers of $K$. In this article, we study the asymptotic solutions of the generalized Fermat equation, i.e., $Ax^p+By^p+Cz^p=0$ over $K$ of prime exponent $p$, where $A,B,C \in \mathcal{O}_K \setminus \{0\}$ with $ABC$ is even (in the sense that $\mathfrak{P}| ABC$, for some prime ideal $\mathfrak{P}$ of $ \mathcal{O}_K$ with $\mathfrak{P} |2$). For certain class of fields $K$, we prove that the equation $Ax^…

Combining Statistical Depth and Fermat Distance for Uncertainty Quantification
Hai-Vy Nguyen, Fabrice Gamboa, Reda Chhaibi, Sixin Zhang, Serge Gratton, Thierry Giaccone
https://arxiv.org/abs/2404.08476

Combining Statistical Depth and Fermat Distance for Uncertainty Quantification
We measure the Out-of-domain uncertainty in the prediction of Neural Networks using a statistical notion called ``Lens Depth'' (LD) combined with Fermat Distance, which is able to capture precisely the ``depth'' of a point with respect to a distribution in feature space, without any assumption about the form of distribution. Our method has no trainable parameter. The method is applicable to any classification model as it is applied directly in feature space at test time and does not intervene i…

Fermat Number Transform Based Chromatic Dispersion Compensation and Adaptive Equalization Algorithm
Siyu Chen, Zheli Liu, Weihao Li, Zihe Hu, Mingming Zhang, Sheng Cui, Ming Tang
https://arxiv.org/abs/2405.04253

Fermat Number Transform Based Chromatic Dispersion Compensation and Adaptive Equalization Algorithm
By introducing the Fermat number transform into chromatic dispersion compensation and adaptive equalization, the computational complexity has been reduced by 68% compared with the con?ventional implementation. Experimental results validate its transmission performance with only 0.8 dB receiver sensitivity penalty in a 75 km-40 GBaud-PDM-16QAM system.

This https://arxiv.org/abs/2402.14287 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Tropical Fermat-Weber Polytopes
We study the geometry of tropical Fermat-Weber points in terms of the symmetric tropical metric over the tropical projective torus. It is well known that a tropical Fermat-Weber point of a given sample is not unique and in this paper we show that the set of all possible Fermat-Weber points forms a polytrope. Then, we introduce the tropical Fermat-Weber gradient and using them, we show that the tropical Fermat-Weber polytrope is a bounded cell of a tropical hyperplane arrangement given by both m…

This https://arxiv.org/abs/2309.01059 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

$L$-function of CM elliptic curves and generalized hypergeometric functions
In this paper, we express special values of the $L$-functions of certain CM elliptic curves which are related to Fermat curves in terms of the special values of generalized hypergeometric functions by comparing Bloch's element with Ross's element in the motivic cohomology group.

This https://arxiv.org/abs/2401.00470 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_hept…

More on $G$-flux and General Hodge Cycles on the Fermat Sextic
We study M-Theory solutions with $G$-flux on the Fermat sextic Calabi-Yau fourfold, focussing on the relationship between the number of stabilized complex structure moduli and the tadpole contribution of the flux. We use two alternative approaches to define the fluxes: algebraic cycles and (appropriately quantized) Griffiths residues. In both cases, we collect evidence for the non-existence of solutions which stabilize all moduli and stay within the tadpole bound

Mechanophotonics: Pseudo-plastic Organic Crystal as a Fermat Spiral Optical Waveguide
Melchi Chosenyah, Avulu Vinod Kumar, Rajadurai Chandrasekar
https://arxiv.org/abs/2402.18878 …

Mechanophotonics: Pseudo-plastic Organic Crystal as a Fermat Spiral Optical Waveguide
An unprecedented organic Fermat spiral optical waveguide (FSOW) self transducing green fluorescence is fabricated using a pseudo-plastic (E)-1-(((5-bromopyridin-2-yl)imino)methyl)naphthalene-2-ol crystal. A 1.618-millimeter-long crystal is initially bent into a hairpin-like bent waveguide. Later, a meticulous mechanophotonic strategy is employed to sculpt the hairpin-like bent waveguide into the Fermat spiral geometry, covering a compact area of 330x238 um2. The optical signal in FSOW survives …

This https://arxiv.org/abs/2210.03744 has been replaced.
link: https://scholar.google.com/scholar?q=a

Fermat's Last Theorem over $\mathbb{Q}(\sqrt{2},\sqrt{3})$
In this paper, we begin the study of the Fermat equation $x^n+y^n=z^n$ over real biquadratic fields. In particular, we prove that there are no non-trivial solutions to the Fermat equation over $\mathbb{Q}(\sqrt{2},\sqrt{3})$ for $n\geq 4$.

This https://arxiv.org/abs/2312.07710 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

The classifying element for quotients of Fermat curves
Suppose $C$ is a cyclic Galois cover of the projective line branched at the three points $0$, $1$, and $\infty$. Under a mild condition on the ramification, we determine the structure of the graded Lie algebra of the lower central series of the fundamental group of $C$ in terms of a basis which is well-suited to studying the action of the absolute Galois group of $\mathbb{Q}$.

Free curves in Fano hypersurfaces must have high degree
Raymond Cheng
https://arxiv.org/abs/2404.17341 https://arxiv.org/pdf/2404.173…

Free curves in Fano hypersurfaces must have high degree
The purpose of this note is to show that the minimal $e$ for which every smooth Fano hypersurface of dimension $n$ contains a free rational curve of degree at most $e$ cannot be bounded by a linear function in $n$ when the base field has positive characteristic. This is done by providing a super-linear bound on the minimal possible degree of a free curve in certain Fermat hypersurfaces.

Points totalement r\'eels de la courbe $x^5 y^5 z^5=0$
Alain Kraus
https://arxiv.org/abs/2404.08922 https://arxiv.org/pdf/2404.08…

Points totalement réels de la courbe $x^5+y^5+z^5=0$
Let $\overline{\mathbb{Q}}$ be an algebraic closure of $\mathbb{Q}$ and $\mathbb{Q}^{tr}$ be the subfield of $\overline{\mathbb{Q}}$ obtained by taking the union of all totally real number fields. For any prime $p\geq 3$, let $F_p/\mathbb{Q}$ be the Fermat curve of equation $x^p+y^p+z^p=0$. In 1996, Pop has shown that the field $\mathbb{Q}^{tr}$ is large. In particular, the set $F_p(\mathbb{Q}^{tr})$ of the points of $F_p$ rational over $\mathbb{Q}^{tr}$ is infinite. How to explicit non-trivial…

A class of explicit solutions for the Fermat problem for tetrahedra
Anastasios N. Zachos
https://arxiv.org/abs/2405.08004 https://arx…

Asymptotic solutions of generalized Fermat-type equation of signature $(p,p,3)$ over totally real number fields
Narasimha Kumar, Satyabrat Sahoo
https://arxiv.org/abs/2403.14640 <…

Asymptotic solutions of generalized Fermat-type equation of signature $(p,p,3)$ over totally real number fields
In this article, we study the asymptotic solutions of the generalized Fermat-type equation of signature $(p,p,3)$ over totally real number fields $K$, i.e., $Ax^p+By^p=Cz^3$ with prime exponent $p$ and $A,B,C \in \mathcal{O}_K \setminus \{0\}$. For certain class of fields $K$, we prove that $Ax^p+By^p=Cz^3$ has no asymptotic solutions over $K$ (resp., solutions of certain type over $K$) with restrictions on $A,B,C$ (resp., for all $A,B,C \in \mathcal{O}_K \setminus \{0\}$). Finally, we present …

Jacob's ladders, Hardy-Littlewood integral (1918) and new asymptotic functional equations for Euler's Gamma function together with the tenth equivalent of the Fermat-Wiles theorem
Jan Moser
https://arxiv.org/abs/2403.17522

Jacob's ladders, Hardy-Littlewood integral (1918) and new asymptotic functional equations for Euler's Gamma function together with the tenth equivalent of the Fermat-Wiles theorem
In this paper new $Γ$-functional is constructed upon the basis of the set of almost linear increments of the Hardy-Littlewood integral. This functional generates a $Γ$-equivalent of the Fermat-Wiles theorem and also new set of factorization formulae for Euler's $Γ$-function.

Refraction, the speed of light and minimal action: From Descartes to Maupertuis through many more
Shahen Hacyan
https://arxiv.org/abs/2402.19400 https://…

Refraction, the speed of light and minimal action: From Descartes to Maupertuis through many more
In the 17th and 18th centuries, several natural philosophers studied the phenomenon of refraction and attempted to obtain the Snell law from various assumptions. Lacking experimental data, it was generally believed that light travels faster in a refracting medium than in air. In the present article, I review the contributions to the problem of light refraction by Descartes, Fermat, Huygens, Leibniz, Newton, Clairaut, and finally Maupertuis who established a principle of least action based on hi…

This https://arxiv.org/abs/2404.12906 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

A Pythagorean triangle in which the hypotenuse and the sum of the arms are squares
In this paper, show that the Diophantine equation $ x^2+(x+1)^2=w^4 $ has only two solutions $ (0,1) $ and $ (119,13)$ in non-negative integers $ x $ and $ w $. This equation concerned a classic problem posed by Pierre de Fermat, wonders about finding a Pythagorean triangle in which the hypotenuse and the sum of the arms are square. We review the method of finding the smallest solution presented by Fermat, and the relationship between the primitive Pythagorean triples and the Pell's equation, F…

Refraction, the speed of light and minimal action: From Descartes to Maupertuis through many more
Shahen Hacyan
https://arxiv.org/abs/2402.19400 https://…

Refraction, the speed of light and minimal action: From Descartes to Maupertuis through many more
In the 17th and 18th centuries, several natural philosophers studied the phenomenon of refraction and attempted to obtain the Snell law from various assumptions. Lacking experimental data, it was generally believed that light travels faster in a refracting medium than in air. In the present article, I review the contributions to the problem of light refraction by Descartes, Fermat, Huygens, Leibniz, Newton, Clairaut, and finally Maupertuis who established a principle of least action based on hi…

This https://arxiv.org/abs/2403.16603 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

On the $\mathbb{Z}_p$-extensions of a totally $p$-adic imaginary quadratic field -- With an appendix by Jean-François Jaulent
Let $k$ be an imaginary quadratic field, and let $p$ be an odd prime number split in $k$. We analyze some properties of arbitrary $\mathbb{Z}_p$-extensions K/k. These properties are governed by the Hase norm residue symbol of the fundamental $p$-unit of $k$, in terms of the valuation $δ_p(k)$ of a Fermat quotient, which determines the order of the logarithmic class group $\widetilde{\mathcal{H}_k}$ (Theorem 2.2) and leads, under some conditions, to generalizations of Gold's criterion character…

This https://arxiv.org/abs/2403.16603 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

On the $\mathbb{Z}_p$-extensions of a totally $p$-adic imaginary quadratic field -- With an appendix by Jean-François Jaulent
Let $k$ be an imaginary quadratic field, and let $p$ be an odd prime number split in $k$. We analyze some properties of arbitrary $\mathbb{Z}_p$-extensions K/k. These properties are governed by the Hase norm residue symbol of the fundamental $p$-unit of $k$, in terms of the valuation $δ_p(k)$ of a Fermat quotient, which determines the order of the logarithmic class group $\widetilde{\mathcal{H}_k}$ (Theorem 2.2) and leads, under some conditions, to generalizations of Gold's criterion character…