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Journal articles on the topic 'Mahler equations'

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Published: 28 September 2022

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1

Bugeaud, Yann, and Kálmán Győry. "On binomial Thue-Mahler equations." Periodica Mathematica Hungarica 49, no. 2 (December 2004): 25–34. http://dx.doi.org/10.1007/s10998-004-0520-0.

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2

Lalín, Matilde N. "Equations for Mahler measure and isogenies." Journal de Théorie des Nombres de Bordeaux 25, no. 2 (2013): 387–99. http://dx.doi.org/10.5802/jtnb.841.

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3

Dreyfus, Thomas, Charlotte Hardouin, and Julien Roques. "Hypertranscendence of solutions of Mahler equations." Journal of the European Mathematical Society 20, no. 9 (June 29, 2018): 2209–38. http://dx.doi.org/10.4171/jems/810.

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4

Chyzak, Frédéric, Thomas Dreyfus, Philippe Dumas, and Marc Mezzarobba. "Computing solutions of linear Mahler equations." Mathematics of Computation 87, no. 314 (July 2, 2018): 2977–3021. http://dx.doi.org/10.1090/mcom/3359.

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5

Nishioka, Kumiko, and Seiji Nishioka. "Autonomous equations of Mahler type and transcendence." Tsukuba Journal of Mathematics 39, no. 2 (March 2016): 251–57. http://dx.doi.org/10.21099/tkbjm/1461270059.

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6

Roques, Julien. "On the reduction modulo $p$ of Mahler equations." Tohoku Mathematical Journal 69, no. 1 (April 2017): 55–65. http://dx.doi.org/10.2748/tmj/1493172128.

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7

Kim, Dohyeong. "A modular approach to cubic Thue-Mahler equations." Mathematics of Computation 86, no. 305 (September 15, 2016): 1435–71. http://dx.doi.org/10.1090/mcom/3139.

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8

Nishioka, Kumiko, and Seiji Nishioka. "Algebraic theory of difference equations and Mahler functions." Aequationes mathematicae 84, no. 3 (May 11, 2012): 245–59. http://dx.doi.org/10.1007/s00010-012-0132-3.

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9

Bugeaud, Yann, and Kálmán Győry. "Bounds for the solutions of Thue-Mahler equations and norm form equations." Acta Arithmetica 74, no. 3 (1996): 273–92. http://dx.doi.org/10.4064/aa-74-3-273-292.

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10

Lalin, Matilde, and Mathew Rogers. "Functional equations for Mahler measures of genus-one curves." Algebra & Number Theory 1, no. 1 (February 1, 2007): 87–117. http://dx.doi.org/10.2140/ant.2007.1.87.

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11

Smart, N. P. "Thue and Thue-Mahler Equations over Rings of Integers." Journal of the London Mathematical Society 56, no. 3 (December 1997): 455–62. http://dx.doi.org/10.1112/s0024610797005619.

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12

Nishioka, Kumiko, and Thomas T�pfer. "Transcendence measures and nonlinear functional equations of Mahler type." Archiv der Mathematik 57, no. 4 (October 1991): 370–78. http://dx.doi.org/10.1007/bf01198962.

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13

Becker, P. G. "k-Regular Power Series and Mahler-Type Functional Equations." Journal of Number Theory 49, no. 3 (December 1994): 269–86. http://dx.doi.org/10.1006/jnth.1994.1093.

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14

Brindza, B., J. H. Evertse, and K. Györy. "Bounds for the solutions of some diophantine equations in terms of discriminants." Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 51, no. 1 (August 1991): 8–26. http://dx.doi.org/10.1017/s1446788700033267.

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AbstractSeveral effective upper bounds are known for the solutions of Thue equations, Thue-Mahler equations and superelliptic equations. One of the basic parameters occurring in these bounds is the height of the polynomial involved in the equation. In the present paper it is shown that better (and, in certain important particular cases, best possible) upper bounds can be obtained in terms of the height, if one takes into consideration also the discriminant of the polynomial.
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15

Amou, Masaaki. "An improvement of a transcendence measure of Galochkin and Mahler's S-numbers." Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 52, no. 1 (February 1992): 130–40. http://dx.doi.org/10.1017/s1446788700032912.

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AbstractWe give a transcendence measure of special values of functions satisfying certain functional equations. This improves an earlier result of Galochkin, and gives a better upper bound of the type for such a number as an S-number in the classification of transcendental numbers by Mahler.
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16

Väänänen, Keijo, and Wen Wu. "On linear independence measures of the values of Mahler functions." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 148, no. 6 (June 22, 2018): 1297–311. http://dx.doi.org/10.1017/s0308210518000148.

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We estimate the linear independence measures for the values of a class of Mahler functions of degrees 1 and 2. For this purpose, we study the determinants of suitable Hermite–Padé approximation polynomials. Based on the non-vanishing property of these determinants, we apply the functional equations to get an infinite sequence of approximations that is used to produce the linear independence measures.
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17

Töpfer, Thomas. "Zero order estimates for functions satisfying generalized functional equations of Mahler type." Acta Arithmetica 85, no. 1 (1998): 1–12. http://dx.doi.org/10.4064/aa-85-1-1-12.

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18

Greuel, Bernd. "Algebraic independence of the values of Mahler functions satisfying implicit functional equations." Acta Arithmetica 93, no. 1 (2000): 1–20. http://dx.doi.org/10.4064/aa-93-1-1-20.

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19

Töpfer, Th. "Simultaneous approximation measures for functions satisfying generalized functional equations of mahler type." Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 66, no. 1 (December 1996): 177–201. http://dx.doi.org/10.1007/bf02940803.

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20

BUNDSCHUH, PETER, and KEIJO VÄÄNÄNEN. "ALGEBRAIC INDEPENDENCE OF CERTAIN MAHLER FUNCTIONS AND OF THEIR VALUES." Journal of the Australian Mathematical Society 98, no. 3 (November 11, 2014): 289–310. http://dx.doi.org/10.1017/s1446788714000524.

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This paper considers algebraic independence and hypertranscendence of functions satisfying Mahler-type functional equations $af(z^{r})=f(z)+R(z)$, where $a$ is a nonzero complex number, $r$ an integer greater than 1, and $R(z)$ a rational function. Well-known results from the scope of Mahler’s method then imply algebraic independence over the rationals of the values of these functions at algebraic points. As an application, algebraic independence results on reciprocal sums of Fibonacci and Lucas numbers are obtained.
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21

BUGEAUD, YANN. "Effective irrationality measures for real and p-adic roots of rational numbers close to 1, with an application to parametric families of Thue–Mahler equations." Mathematical Proceedings of the Cambridge Philosophical Society 164, no. 1 (September 27, 2016): 99–108. http://dx.doi.org/10.1017/s0305004116000864.

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AbstractWe show how the theory of linear forms in two logarithms allows one to get very good effective irrationality measures for nth roots of rational numbers a/b, when a is very close to b. We give a p-adic analogue of this result under the assumption that a is p-adically very close to b, that is, that a large power of p divides a−b. As an application, we solve completely certain families of Thue–Mahler equations. Our results illustrate, admittedly in a very special situation, the strength of the known estimates for linear forms in logarithms.
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22

Toledano, R. "The Mahler Measure of Linear Forms as Special Values of Solutions of Algebraic Differential Equations." Rocky Mountain Journal of Mathematics 39, no. 4 (August 2009): 1323–38. http://dx.doi.org/10.1216/rmj-2009-39-4-1323.

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23

EVERTSE, JAN–HENDRIK, and KÁLMÁN GYŐRY. "Effective results for unit equations over finitely generated integral domains." Mathematical Proceedings of the Cambridge Philosophical Society 154, no. 2 (November 23, 2012): 351–80. http://dx.doi.org/10.1017/s0305004112000606.

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AbstractLet A ⊃ ℤ be an integral domain which is finitely generated over ℤ and let a,b,c be non-zero elements of A. Extending earlier work of Siegel, Mahler and Parry, in 1960 Lang proved that the equation (*) aϵ +bη = c in ϵ, η ∈ A* has only finitely many solutions. Using Baker's theory of logarithmic forms, Győry proved, in 1979, that the solutions of (*) can be determined effectively if A is contained in an algebraic number field. In this paper we prove, in a quantitative form, an effective finiteness result for equations (*) over an arbitrary integral domain A of characteristic 0 which is finitely generated over ℤ. Our main tools are already existing effective finiteness results for (*) over number fields and function fields, an effective specialization argument developed by Győry in the 1980's, effective results of Hermann (1926) and Seidenberg (1974) on linear equations over polynomial rings over fields, and similar such results by Aschenbrenner, from 2004, on linear equations over polynomial rings over ℤ. We prove also an effective result for the exponential equation aγ1v1···γsvs+bγ1w1 ··· γsws=c in integers v1,…,ws, where a,b,c and γ1,…,γs are non-zero elements of A.
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24

ZORIN, EVGENIY. "ZERO ORDER ESTIMATES FOR ANALYTIC FUNCTIONS." International Journal of Number Theory 09, no. 02 (December 5, 2012): 333–92. http://dx.doi.org/10.1142/s1793042112501370.

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In this article we develop an important tool in transcendental number theory. More precisely, we study multiplicity estimates (or multiplicity lemmas) for analytic functions. Our main theorem reduces multiplicity estimates at zero to the study of ideals in polynomial ring stable under an appropriate map. In particular, in the case of algebraic morphisms this result gives a new link between the theory of polarized algebraic dynamical systems and transcendental number theory. Specialized to the case of differential operators this theorem improves Nesterenko's conditional result on solutions of systems of differential equations. We also deduce an analog of Nesterenko's theorem for Mahler's functions and for solutions of q-difference equations. Further, analyzing stable ideals we prove the unconditional optimal result in the case of linear functional systems of generalized Mahler's type. The latter result generalizes a famous theorem of Nishioka (1986) previously conjectured by Mahler (1969). This new multiplicity estimate allows to prove new results on algebraic independence and on measures of algebraic independence, as done in Zorin (2010 and 2011).
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25

Goto, Akinari, and Taka-aki Tanaka. "Algebraic independence of the values of functions satisfying Mahler type functional equations under the transformation represented by a power relatively prime to the characteristic of the base field." Journal of Number Theory 184 (March 2018): 384–410. http://dx.doi.org/10.1016/j.jnt.2017.08.026.

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26

Ghioca, Dragos, and Junyi Xie. "The Dynamical Mordell–Lang Conjecture for Skew-Linear Self-Maps. Appendix by Michael Wibmer." International Mathematics Research Notices 2020, no. 21 (September 7, 2018): 7433–53. http://dx.doi.org/10.1093/imrn/rny211.

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Abstract Let $k$ be an algebraically closed field of characteristic $0$, let $N\in{\mathbb{N}}$, let $g:{\mathbb{P}}^1{\longrightarrow } {\mathbb{P}}^1$ be a nonconstant morphism, and let $A:{\mathbb{A}}^N{\longrightarrow } {\mathbb{A}}^N$ be a linear transformation defined over $k({\mathbb{P}}^1)$, that is, for a Zariski-open dense subset $U\subset{\mathbb{P}}^1$, we have that for $x\in U(k)$, the specialization $A(x)$ is an $N$-by-$N$ matrix with entries in $k$. We let $f:{\mathbb{P}}^1\times{\mathbb{A}}^N{\dashrightarrow } {\mathbb{P}}^1\times{\mathbb{A}}^N$ be the rational endomorphism given by $(x,y)\mapsto (\,g(x), A(x)y)$. We prove that if $g$ induces an automorphism of ${\mathbb{A}}^1\subset{\mathbb{P}}^1$, then each irreducible curve $C\subset{\mathbb{A}}^1\times{\mathbb{A}}^N$ that intersects some orbit $\mathcal{O}_f(z)$ in infinitely many points must be periodic under the action of $f$. Furthermore, in the case $g:{\mathbb{P}}^1{\longrightarrow } {\mathbb{P}}^1$ is an endomorphism of degree greater than $1$, then we prove that each irreducible subvariety $Y\subset{\mathbb{P}}^1\times{\mathbb{A}}^N$ intersecting an orbit $\mathcal{O}_f(z)$ in a Zariski dense set of points must be periodic. Our results provide the desired conclusion in the Dynamical Mordell–Lang Conjecture in a couple new instances. Moreover, our results have interesting consequences toward a conjecture of Rubel and toward a generalized Skolem–Mahler–Lech problem proposed by Wibmer in the context of difference equations. In the appendix it is shown that the results can also be used to construct Picard–Vessiot extensions in the ring of sequences.
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27

Maher, T., A. Bourdin, E. Volkmann, S. Vettori, J. H. W. Distler, M. Alves, C. Stock, and O. Distler. "POS0385 “EFFECTIVE LUNG AGE” IN SUBJECTS WITH SYSTEMIC SCLEROSIS-ASSOCIATED INTERSTITIAL LUNG DISEASE (SSc-ILD) IN THE SENSCIS TRIAL." Annals of the Rheumatic Diseases 81, Suppl 1 (May 23, 2022): 447.1–447. http://dx.doi.org/10.1136/annrheumdis-2022-eular.746.

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BackgroundFVC declines gradually with aging. The loss of lung function in patients with progressive SSc-ILD mimics an accelerated ageing process.ObjectivesUsing reference equations, we estimated the “effective lung age” of subjects in the SENSCIS trial (i.e. the age of healthy individuals with the same FVC) and compared these estimates to their real age.MethodsThe SENSCIS trial enrolled subjects with SSc-ILD with first non-Raynaud symptom within the prior ≤7 years, extent of fibrotic ILD on HRCT ≥10%, FVC ≥40% predicted, DLco 30–89% predicted. Evidence of recent decline in FVC was not an inclusion criterion. Subjects were randomised to receive nintedanib or placebo. Using reference equations published by the European Respiratory Society Global Lung Function Initiative, based on FVC data from over 70,000 healthy individuals aged 3–95 years from 26 countries, [Quanjer et al. Eur Respir J 2012;40:1324–1343], we estimated the effective lung age of subjects at baseline and at week 52 based on their FVC, sex, ethnicity and height, and compared these effective lung ages with the subjects’ real ages. Three subjects aged <25 years were excluded. The upper limit of effective lung age was considered to be 95 years.ResultsMean time since onset of first non-Raynaud symptom was 3.5 years in both the nintedanib and placebo groups. At baseline, mean (SD) effective lung age was 83.1 (14.4) years in the nintedanib group (n=287) and 82.9 (14.8) years in the placebo group (n=286). In these groups, respectively, the mean (SD) difference between effective lung age and real age was 28.4 (17.7) and 29.3 (18.5) years and the difference was >20 years in 71.4% and 72.4% of subjects. In the nintedanib and placebo groups, respectively, median (Q1, Q3) effective lung age was 88.4 (74.6, 95.0) and 88.5 (74.7, 95.0) years at baseline and 91.0 (75.2, 95.0) and 95.0 (75.9, 95.0) years at week 52.ConclusionAt entry into the SENSCIS trial, subjects with SSc-ILD had an effective lung age that was much higher than their real age. Over 52 weeks, the increase in effective lung age was numerically lower in subjects treated with nintedanib than placebo. These data show that marked loss of lung function that can occur in the few years following onset of SSc-ILD and support a benefit of nintedanib in slowing the progression of SSc-ILD.AcknowledgementsThe SENSCIS trial was funded by Boehringer Ingelheim. Toby M Maher and Oliver Distler were members of the SENSCIS trial Steering Committee.Disclosure of InterestsToby Maher Speakers bureau: Boehringer Ingelheim, Galapagos, Genentech, Consultant of: AstraZeneca, Bayer, Blade Therapeutics, Boehringer Ingelheim, Bristol-Myers Squibb, Galapagos, Galecto, GlaxoSmithKline R&D, IQVIA, Pliant, Respivant, Roche, Theravance and Veracyte, Grant/research support from: AstraZeneca, GlaxoSmithKline, Arnaud Bourdin Speakers bureau: Amgen, AstraZeneca, Boehringer Ingelheim, Chiesi, GlaxoSmithKline, Novartis, Regeneron, Sanofi, Paid instructor for: Amgen, AstraZeneca, Boehringer Ingelheim, Chiesi, GlaxoSmithKline, Novartis, Regeneron, Sanofi, Consultant of: Amgen, AstraZeneca, Boehringer Ingelheim, Chiesi, GlaxoSmithKline, Novartis, Regeneron, Sanofi, Grant/research support from: AstraZeneca and Boehringer Ingelheim, Elizabeth Volkmann Speakers bureau: Boehringer Ingelheim, Consultant of: Boehringer Ingelheim, Grant/research support from: Boehringer Ingelheim, Corbus, Forbius, Horizon, Kadmon, Serena Vettori Consultant of: Boehringer Ingelheim, Jörg H.W. Distler Shareholder of: 4D Science, Speakers bureau: Boehringer Ingelheim, Inventiva, Janssen, and UCB, Consultant of: AbbVie, Active Biotech, Anamar, ARXX, AstraZeneca, Bayer Pharma, Boehringer Ingelheim, Celgene, Galapagos, GlaxoSmithKline, Inventiva, Janssen, Novartis, Pfizer, and UCB, Grant/research support from: Anamar, ARXX, Bristol-Myers Squibb, Bayer Pharma, Boehringer Ingelheim, Cantargia, Celgene, CSL Behring, Galapagos, GlaxoSmithKline, Inventiva, Kiniksa, Sanofi-Aventis, RedX, UCB, Margarida Alves Employee of: Margarida Alves is an employee of Boehringer Ingelheim, Christian Stock Employee of: Christian Stock is an employee of Boehringer Ingelheim, Oliver Distler Speakers bureau: OD has/had relationships with the following companies in the area of potential treatments for systemic sclerosis and its complications in the last three calendar years:Speaker fee: Bayer, Boehringer Ingelheim, Janssen, Medscape, Consultant of: OD has/had relationships with the following companies in the area of potential treatments for systemic sclerosis and its complications in the last three calendar years:Consultancy fee: Abbvie, Acceleron, Alcimed, Amgen, AnaMar, Arxx, AstraZeneca, Baecon, Blade, Bayer, Boehringer Ingelheim, Corbus, CSL Behring, 4P Science, Galapagos, Glenmark, Horizon, Inventiva, Kymera, Lupin, Miltenyi Biotec, Mitsubishi Tanabe, MSD, Novartis, Prometheus, Roivant, Sanofi and TopadurOD has/had relationships with the following companies in the area of potential treatments for arthritides in the last three calendar years:Consultancy fee: Abbvie, Grant/research support from: OD has/had relationships with the following companies in the area of potential treatments for systemic sclerosis and its complications in the last three calendar years:Research Grants: Boehringer Ingelheim, Kymera, Mitsubishi Tanabe
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28

Maher, T., A. Bourdin, E. Volkmann, S. Vettori, J. H. W. Distler, M. Alves, C. Stock, and O. Distler. "POS0835 DECLINE IN FORCED VITAL CAPACITY (FVC) IN SUBJECTS WITH SYSTEMIC SCLEROSIS-ASSOCIATED INTERSTITIAL LUNG DISEASE (SSC-ILD) IN THE SENSCIS TRIAL VERSUS HYPOTHETICAL REFERENCE SUBJECTS WITHOUT LUNG DISEASE." Annals of the Rheumatic Diseases 80, Suppl 1 (May 19, 2021): 671–72. http://dx.doi.org/10.1136/annrheumdis-2021-eular.897.

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Background:In the randomized SENSCIS trial in subjects with SSc-ILD, nintedanib reduced the rate of decline in FVC over 52 weeks (mL/year) by 44% compared to placebo. Healthy individuals have varied FVC depending on age, sex, ethnicity and height; expected values can be determined using internationally recognised reference equations.Objectives:To provide further context to the FVC declines observed in the SENSCIS trial, we compared the decline in FVC observed in subjects with SSc-ILD in the SENSCIS trial with the decline in FVC that would be expected in hypothetical subjects without ILD matched for age, sex, ethnicity and height.Methods:The SENSCIS trial enrolled subjects with SSc-ILD aged ≥18 years with first non-Raynaud symptom ≤7 years before screening, extent of fibrotic ILD ≥10% on HRCT, FVC ≥40% predicted and DLco 30–89% predicted. Baseline FVC (mL) and changes in FVC (mL) at week 52 were assessed in the nintedanib and placebo groups, with missing values at week 52 imputed using predictions from the primary analysis model (random slope and intercept model). Changes in FVC in the SENSCIS trial were compared to values in hypothetical healthy reference subjects matched to the SENSCIS subjects for age, sex, ethnicity and height. FVC values in these healthy reference subjects were derived from the equations published by the European Respiratory Society Global Lung Function Initiative in 2012, which were derived from data from over 70,000 subjects.1Results:In the nintedanib and placebo groups of the SENSCIS trial, respectively, mean (SD) time since onset of first non-Raynaud symptom was 3.5 (1.6) and 3.5 (1.8) years. In the nintedanib group, mean (SD) FVC at baseline was 2460 (737) mL, compared with 3403 (787) mL in the healthy reference subjects. In the placebo group, mean (SD) FVC at baseline was 2544 (817) mL compared with 3516 (887) mL in the healthy reference subjects. The difference in the change from baseline in FVC at week 52 between the nintedanib-treated subjects in the SENSCIS trial (n=287) and the healthy reference subjects was 26.6 mL ([95% CI: 1.2, 52.0]; p=0.04). The difference in the change from baseline in FVC at week 52 between the placebo-treated subjects in the SENSCIS trial (n=286) and the reference subjects was 77.5 mL ([95% CI: 51.4, 103.7]; p<0.001) (Figure 1).Conclusion:Subjects with SSc-ILD who participated in the SENSCIS trial had marked lung function impairment at baseline compared with healthy matched reference subjects, despite a mean duration of SSc of 3.5 years. Over 52 weeks, the decline in FVC in subjects with SSc-ILD who received placebo was 4-fold greater than in healthy reference subjects. Subjects with SSc-ILD who were treated with nintedanib had a decline in FVC that was only slightly greater than the decline observed in the matched healthy subjects. These data support the clinical relevance of the reduction in the rate of FVC decline provided by nintedanib in patients with SSc-ILD.References:[1]Quanjer et al. Eur Respir J 2012;40:1324−43.Acknowledgements:The SENSCIS trial was funded by Boehringer Ingelheim. Medical writing support was provided by FleishmanHillard Fishburn, London, UK. The authors meet criteria for authorship as recommended by the International Committee of Medical Journal Editors (ICMJE).Disclosure of Interests:Toby Maher Speakers bureau: Boehringer Ingelheim and Roche/Genentech, Consultant of: Acelleron Pharma, AstraZeneca, Boehringer Ingelheim, Bristol-Myers Squibb, GlaxoSmithKline and Roche/Genentech, Arnaud Bourdin Speakers bureau: Actelion/Janssen (personal fees and other), AstraZeneca (personal fees and other), Boeringher Ingelheim (personal fees and other), Chiesi (personal fees and other), GlaxoSmithKline (personal fees and other), Novartis (personal fees and other), Pulsar Therapeutics (other), Roche (personal fees and other), Sanofi Regeneron (personal fees and other), Teva (other) and United Therapeutics (other), Consultant of: Actelion/Janssen (personal fees and other), AstraZeneca (personal fees and other), Boeringher Ingelheim (personal fees and other), Chiesi (personal fees and other), GlaxoSmithKline (personal fees and other), Novartis (personal fees and other), Pulsar Therapeutics (other), Roche (personal fees and other), Sanofi Regeneron (personal fees and other), Teva (other) and United Therapeutics (other), Grant/research support from: Actelion/Janssen (grants and other), AstraZeneca (grants and other), Boeringher Ingelheim (grants and other), Chiesi (other), GlaxoSmithKline (grants and other), Novartis (other), Pulsar Therapeutics (other), Roche (other), Sanofi Regeneron (other), Teva (other) and United Therapeutics (other), Elizabeth Volkmann Consultant of: Boehringer Ingelheim, Grant/research support from: Corbus and Forbius, Serena Vettori Paid instructor for: Boehringer Ingelheim, Consultant of: Boehringer Ingelheim, Jörg H.W. Distler Speakers bureau: Actelion, Active Biotech, AnaMar, Arxx Therapeutics, Bayer, Boehringer Ingelheim, Celgene, Galapagos NV, GlaxoSmithKline, Inventiva, JB Therapeutics, Medac, Pfizer, RuiYi and UCB, Consultant of: AnaMar, Arxx Therapeutics, Bayer, Boehringer Ingelheim, Galapagos NV, Inventiva, JB Therapeutics and UCB, Grant/research support from: Active Biotech, AnaMar, Array BioPharma, Arxx Therapeutics, aTyr, Bayer, Boehringer Ingelheim, Bristol-Myers Squibb, Celgene, Galapagos NV, GlaxoSmithKline, Inventiva, Novartis, Sanofi-Aventis, Redx and UCB, Margarida Alves Employee of: Currently an employee of Boehringer Ingelheim, Christian Stock Employee of: Currently an employee of Boehringer Ingelheim, Oliver Distler Consultant of: AbbVie, Acceleron Pharma, Amgen, AnaMar, Arxx Therapeutics, Baecon Discovery, Bayer, Blade Therapeutics, Boehringer Ingelheim, ChemomAb, Corbus, CSL Behring, Galapagos NV, GlaxoSmithKline, Glenmark Pharmaceuticals, Horizon (Curzion) Pharmaceuticals, Inventiva, IQVIA, Italfarmaco, iQone, Kymera Therapeutics, Lilly, Medac, Medscape, Merck Sharp & Dohme, Mitsubishi Tanabe Pharma, Novartis, Pfizer, Roche, Sanofi, Serodapharm, Target Bioscience, Topadur Pharma and UCB, Grant/research support from: Kymera Therapeutics and Mitsubishi Tanabe Pharma
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29

BUGEAUD, YANN, CLAUDE LEVESQUE, and MICHEL WALDSCHMIDT. "Equations de Fermat--Pell--Mahler simultanees." Publicationes Mathematicae Debrecen, December 1, 2011, 357–66. http://dx.doi.org/10.5486/pmd.2011.5192.

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30

"Thue-Mahler equations with a small number of solutions." Journal für die reine und angewandte Mathematik (Crelles Journal) 1989, no. 399 (August 1, 1989): 60–80. http://dx.doi.org/10.1515/crll.1989.399.60.

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31

Roques, Julien. "On the Local Structure of Mahler Systems." International Mathematics Research Notices, January 18, 2020. http://dx.doi.org/10.1093/imrn/rnz349.

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Abstract This paper is a 1st step in the direction of a better understanding of the structure of the so-called Mahler systems: we classify these systems over the field $\mathscr{H}$ of Hahn series over $\overline{{\mathbb{Q}}}$ and with value group ${\mathbb{Q}}$. As an application of (a variant of) our main result, we give an alternative proof of the following fact: if, for almost all primes $p$, the reduction modulo $p$ of a given Mahler equation with coefficients in ${\mathbb{Q}}(z)$ has a full set of algebraic solutions over $\mathbb{F}_{p}(z)$, then the given equation has a full set of solutions in $\overline{{\mathbb{Q}}}(z)$ (this is analogous to Grothendieck’s conjecture for differential equations).
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"Transcendence of the values of functions satisfying generalized Mahler type functional equations." Journal für die reine und angewandte Mathematik (Crelles Journal) 1993, no. 440 (July 1, 1993): 111–28. http://dx.doi.org/10.1515/crll.1993.440.111.

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