Rounding and Chaining LLL: Finding Faster Small Roots of Univariate Polynomial Congruences

We will determine the number of powers ofαthat appear with nonzero coefficient in anα-power linear differential resolvent of smallest possible order of a univariate polynomialP(t)whose coefficients lie in an ordinary differential field and whose distinct roots are differentially independent over constants. We will then give an upper bound on the weight of anα-resolvent of smallest possible weight. We will then compute the indicial equation, apparent singularities, and Wronskian of the Cockleα-resolvent of a trinomial and finish with a related determinantal formula.

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SIMULTANEOUS COMPUTATION OF ALL THE ZERO-CLUSTERS OF A UNIVARIATE POLYNOMIAL

Foundations of Computational Mathematics ◽

10.1142/9789812778031_0018 ◽

2002 ◽

Cited By ~ 2

Author(s):

JEAN-CLAUDE YAKOUBSOHN

Keyword(s):

Univariate Polynomial

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The approximate irreducible factorization of a univariate polynomial

Proceedings of the 2009 international symposium on Symbolic and algebraic computation - ISSAC '09 ◽

10.1145/1576702.1576752 ◽

2009 ◽

Cited By ~ 2

Author(s):

Zhonggang Zeng

Keyword(s):

Irreducible Factorization ◽

Univariate Polynomial

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How many integer homogeneous polynomials at small coprime integers have value of a univariate polynomial?

Lithuanian Mathematical Journal ◽

10.1007/s10986-012-9188-y ◽

2012 ◽

Vol 52 (4) ◽

pp. 477-487

Author(s):

Albertas Zinevičius

Keyword(s):

Homogeneous Polynomials ◽

Coprime Integers ◽

Univariate Polynomial

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A general approach to -optimal designs for weighted univariate polynomial regression models

Journal of the Korean Statistical Society ◽

10.1016/j.jkss.2009.11.002 ◽

2010 ◽

Vol 39 (1) ◽

pp. 1-26 ◽

Cited By ~ 7

Author(s):

Holger Dette ◽

Matthias Trampisch

Keyword(s):

Regression Models ◽

Polynomial Regression ◽

Optimal Designs ◽

Univariate Polynomial ◽

Polynomial Regression Models

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Water transport in European Beech roots (<5mm) under drought in the south of Lower Saxony

10.5194/egusphere-egu21-2859 ◽

2021 ◽

Author(s):

Eva Messinger ◽

Heinz Coners ◽

Dietrich Hertel ◽

Christoph Leuschner

Keyword(s):

Soil Moisture ◽

Water Transport ◽

Water Uptake ◽

Sap Flow ◽

Climate Models ◽

European Beech ◽

Root Surface Area ◽

Lower Saxony ◽

Tree Roots ◽

Small Roots

Climate models predict hotter and dryer summers in Germany, with longer periods of extreme droughts like in summer 2018. How does this affect the water uptake and transport in tree roots growing in the top- and subsoil?In summer 2018 and 2019 we measured the water transport in fine roots (<5mm) of European Beech on tertiary sand and triassic sandstone up to 2 m depth. We adapted the well-established HRM technique to enable measurements of very small sap flow rates in small roots. Thus, we measured the water transport as a temperature ratio of a stretching heat pulse.Relating sap flow to root surface area, root depth, anatomy, soil moisture, and VPD allows for interesting insights in tree water uptake rates: Where are the limits of drought intensity and duration, for water uptake and recovery of small roots? Are there differences in the function of top- and subsoil roots? Are roots specialized for water transport or nutrient uptake? The investigated data gives a first hint on how the water transport in Beech roots differs with changes in the soil moisture and VPD under changing climate.

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Hyperbolic Polynomials and Convex Analysis

Canadian Journal of Mathematics ◽

10.4153/cjm-2001-020-6 ◽

2001 ◽

Vol 53 (3) ◽

pp. 470-488 ◽

Cited By ~ 28

Author(s):

Heinz H. Bauschke ◽

Osman Güler ◽

Adrian S. Lewis ◽

Hristo S. Sendov

Keyword(s):

Convex Function ◽

Interior Point ◽

Convex Analysis ◽

Symmetric Functions ◽

Interior Point Methods ◽

Optimization Problems ◽

Real Polynomial ◽

Hyperbolic Polynomials ◽

Real Roots ◽

Univariate Polynomial

AbstractA homogeneous real polynomial p is hyperbolic with respect to a given vector d if the univariate polynomial t ⟼ p(x − td) has all real roots for all vectors x. Motivated by partial differential equations, Gårding proved in 1951 that the largest such root is a convex function of x, and showed various ways of constructing new hyperbolic polynomials. We present a powerful new such construction, and use it to generalize Gårding’s result to arbitrary symmetric functions of the roots. Many classical and recent inequalities follow easily. We develop various convex-analytic tools for such symmetric functions, of interest in interior-point methods for optimization problems over related cones.

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ON SOLUTIONS TO SOME POLYNOMIAL CONGRUENCES IN SMALL BOXES

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972713000671 ◽

2013 ◽

Vol 89 (2) ◽

pp. 300-307

Author(s):

IGOR E. SHPARLINSKI

Keyword(s):

Exponential Sums ◽

Polynomial Systems ◽

Side Length ◽

Character Sum ◽

Number Of Solutions ◽

Polynomial Congruences

AbstractWe use bounds of mixed character sum to study the distribution of solutions to certain polynomial systems of congruences modulo a prime $p$. In particular, we obtain nontrivial results about the number of solutions in boxes with the side length below ${p}^{1/ 2} $, which seems to be the limit of more general methods based on the bounds of exponential sums along varieties.

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