Estimate of a multidimensional sum with the Legendre symbol for a polynomial of odd degree

SynopsisWe show that if B(z) is either (i) a transcendental entire function with order (B)≠1, or (ii) a polynomial of odd degree, then every solution f≠0 to the equation f″ + e−zf′ + B(z)f = 0 has infinite order. We obtain a partial result in the case when B(z) is an even degree polynomial. Our method of proof and lemmas for case (i) of the above result have independent interest.

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Method for assessing single indicators of quality of buildings and structures at hazardous production facilities using S-shaped curves

Quality Innovation Education ◽

10.31145/1999-513x-2021-5-70-74 ◽

2021 ◽

pp. 70-74

Author(s):

M.Yu. Narkevich ◽

Keyword(s):

Quality Assessment ◽

Least Squares Method ◽

Nonlinear Dependence ◽

Quality Estimation ◽

Single Indicator ◽

Odd Degree ◽

Hazardous Production ◽

Polynomial Of Odd Degree ◽

Production Facilities

The article considers the issue of establishing a mathematical dependence in the form of an S-shaped curve for quantifying the quality of buildings and structures at hazardous production facilities with subsequent automation of the quality assessment calculation mechanism based on the obtained mathematical model. It is proposed to take a piecewise nonlinear dependence described by a polynomial of odd degree as the basis of an S-shaped curve. Using the least squares method, the equation of the S-shaped curve of the quality assessment M from the value of a single quality indicator p_(i ) is obtained. The software «Qualimetric Unit Quality Estimation» has been developed, which allows to carry out a quantitative assessment of the quality of a single indicator by the method of qualimetry, as well as to give a direct visual representation in the form of a graph. The process of calculating the quality assessment of the selected single indicator reduces the time for calculating and analyzing the results, increases the efficiency of the procedure for assessing the quality of materials, products, structures of buildings and structures at hazardous production facilities.

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Waring’s Problem for Rational Functions in One Variable

The Quarterly Journal of Mathematics ◽

10.1093/qmathj/haz052 ◽

2020 ◽

Vol 71 (2) ◽

pp. 439-449

Author(s):

Bo-Hae Im ◽

Michael Larsen

Keyword(s):

Rational Function ◽

Sufficient Conditions ◽

Rational Functions ◽

Laurent Polynomial ◽

Necessary And Sufficient Conditions ◽

Waring’S Problem ◽

Waring's Problem ◽

Odd Degree ◽

Necessary And Sufficient ◽

Polynomial Of Odd Degree

Abstract Let $f\in{\mathbb{Q}}(x)$ be a non-constant rational function. We consider ‘Waring’s problem for $f(x)$,’ i.e., whether every element of ${\mathbb{Q}}$ can be written as a bounded sum of elements of $\{f(a)\mid a\in{\mathbb{Q}}\}$. For rational functions of degree $2$, we give necessary and sufficient conditions. For higher degrees, we prove that every polynomial of odd degree and every odd Laurent polynomial satisfies Waring’s problem. We also consider the ‘easier Waring’s problem’: whether every element of ${\mathbb{Q}}$ can be represented as a bounded sum of elements of $\{\pm f(a)\mid a\in{\mathbb{Q}}\}$.

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ON INTEGRAL POINTS ON ISOTRIVIAL ELLIPTIC CURVES OVER FUNCTION FIELDS

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972720000155 ◽

2020 ◽

Vol 102 (2) ◽

pp. 177-185

Author(s):

RICARDO CONCEIÇÃO

Keyword(s):

Elliptic Curve ◽

Elliptic Curves ◽

Function Field ◽

Hyperelliptic Curve ◽

Integral Point ◽

Integral Points ◽

Small Height ◽

Odd Degree ◽

Weil Group ◽

Polynomial Of Odd Degree

Let $k$ be a finite field and $L$ be the function field of a curve $C/k$ of genus $g\geq 1$. In the first part of this note we show that the number of separable $S$-integral points on a constant elliptic curve $E/L$ is bounded solely in terms of $g$ and the size of $S$. In the second part we assume that $L$ is the function field of a hyperelliptic curve $C_{A}:s^{2}=A(t)$, where $A(t)$ is a square-free $k$-polynomial of odd degree. If $\infty$ is the place of $L$ associated to the point at infinity of $C_{A}$, then we prove that the set of separable $\{\infty \}$-points can be bounded solely in terms of $g$ and does not depend on the Mordell–Weil group $E(L)$. This is done by bounding the number of separable integral points over $k(t)$ on elliptic curves of the form $E_{A}:A(t)y^{2}=f(x)$, where $f(x)$ is a polynomial over $k$. Additionally, we show that, under an extra condition on $A(t)$, the existence of a separable integral point of ‘small’ height on the elliptic curve $E_{A}/k(t)$ determines the isomorphism class of the elliptic curve $y^{2}=f(x)$.

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Odd-degree elliptic-function lower-sideband polylithic crystal filters

IEE Proceedings G Circuits Devices and Systems ◽

10.1049/ip-g-2.1993.0014 ◽

1993 ◽

Vol 140 (2) ◽

pp. 91 ◽

Cited By ~ 1

Author(s):

S.K.S. Lu

Keyword(s):

Elliptic Function ◽

Odd Degree

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A Geometric Sequence Binarized with Legendre Symbol over Odd Characteristic Field and Its Properties

IEICE Transactions on Fundamentals of Electronics Communications and Computer Sciences ◽

10.1587/transfun.e97.a.2336 ◽

2014 ◽

Vol E97.A (12) ◽

pp. 2336-2342 ◽

Cited By ~ 25

Author(s):

Yasuyuki NOGAMI ◽

Kazuki TADA ◽

Satoshi UEHARA

Keyword(s):

Legendre Symbol ◽

Geometric Sequence ◽

Characteristic Field

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Optimized CSIDH Implementation Using a 2-Torsion Point

Cryptography ◽

10.3390/cryptography4030020 ◽

2020 ◽

Vol 4 (3) ◽

pp. 20 ◽

Cited By ~ 1

Author(s):

Donghoe Heo ◽

Suhri Kim ◽

Kisoon Yoon ◽

Young-Ho Park ◽

Seokhie Hong

Keyword(s):

Elliptic Curve ◽

Group Action ◽

Large Degree ◽

Transitive Group ◽

Optimization Method ◽

Torsion Point ◽

Torsion Points ◽

Diffie Hellman ◽

Odd Degree ◽

Montgomery Curve

The implementation of isogeny-based cryptography mainly use Montgomery curves, as they offer fast elliptic curve arithmetic and isogeny computation. However, although Montgomery curves have efficient 3- and 4-isogeny formula, it becomes inefficient when recovering the coefficient of the image curve for large degree isogenies. Because the Commutative Supersingular Isogeny Diffie-Hellman (CSIDH) requires odd-degree isogenies up to at least 587, this inefficiency is the main bottleneck of using a Montgomery curve for CSIDH. In this paper, we present a new optimization method for faster CSIDH protocols entirely on Montgomery curves. To this end, we present a new parameter for CSIDH, in which the three rational two-torsion points exist. By using the proposed parameters, the CSIDH moves around the surface. The curve coefficient of the image curve can be recovered by a two-torsion point. We also proved that the CSIDH while using the proposed parameter guarantees a free and transitive group action. Additionally, we present the implementation result using our method. We demonstrated that our method is 6.4% faster than the original CSIDH. Our works show that quite higher performance of CSIDH is achieved while only using Montgomery curves.

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