Tame Decompositions and Collisions

A univariate polynomial f over a field is decomposable if f = g o h = g(h) for nonlinear polynomials g and h. It is intuitively clear that the decomposable polynomials form a small minority among all polynomials over a finite field. The tame case, where the characteristic p of Fq does not divide n = deg f, is fairly well-understood, and we have reasonable bounds on the number of decomposables of degree n. Nevertheless, no exact formula is known if $n$ has more than two prime factors. In order to count the decomposables, one wants to know, under a suitable normalization, the number of collisions, where essentially different (g, h) yield the same f. In the tame case, Ritt's Second Theorem classifies all 2-collisions. We introduce a normal form for multi-collisions of decompositions of arbitrary length with exact description of the (non)uniqueness of the parameters. We obtain an efficiently computable formula for the exact number of such collisions at degree n over a finite field of characteristic coprime to p. This leads to an algorithm for the exact number of decomposable polynomials at degree n over a finite field Fq in the tame case

Survey on counting special types of polynomials

Most integers are composite and most univariate polynomials over a finite field are reducible. The Prime Number Theorem and a classical result of Gau{\ss} count the remaining ones, approximately and exactly. For polynomials in two or more variables, the situation changes dramatically. Most multivariate polynomials are irreducible. This survey presents counting results for some special classes of multivariate polynomials over a finite field, namely the the reducible ones, the s-powerful ones (divisible by the s-th power of a nonconstant polynomial), the relatively irreducible ones (irreducible but reducible over an extension field), the decomposable ones, and also for reducible space curves. These come as exact formulas and as approximations with relative errors that essentially decrease exponentially in the input size. Furthermore, a univariate polynomial f is decomposable if f = g o h for some nonlinear polynomials g and h. It is intuitively clear that the decomposable polynomials form a small minority among all polynomials. The tame case, where the characteristic p of Fq does not divide n = deg f, is fairly well-understood, and we obtain closely matching upper and lower bounds on the number of decomposable polynomials. In the wild case, where p does divide n, the bounds are less satisfactory, in particular when p is the smallest prime divisor of n and divides n exactly twice. The crux of the matter is to count the number of collisions, where essentially different (g, h) yield the same f. We present a classification of all collisions at degree n = p^2 which yields an exact count of those decomposable polynomials.Comment: to appear in Jaime Gutierrez, Josef Schicho & Martin Weimann (editors), Computer Algebra and Polynomials, Lecture Notes in Computer Scienc

Counting classes of special polynomials

Most integers are composite and most univariate polynomials over a finite field are reducible. The Prime Number Theorem and a classical result of Gauß count the remaining ones, approximately and exactly. In two or more variables, the situation changes dramatically. Most multivariate polynomials are irreducible. We present counting results for some special classes of multivariate polynomials over a finite field, namely the reducible ones, the s-powerful ones (divisible by the s-th power of a nonconstant polynomial), and the relatively irreducible ones (irreducible but reducible over an extension field). These numbers come as exact formulas and as approximations with relative errors that essentially decrease exponentially in the input size. Furthermore, a univariate polynomial f over a field F is decomposable if f = g o h with nonlinear polynomials g and h. It is intuitively clear that the decomposable polynomials form a small minority among all polynomials. The tame case, where the characteristic p of F does not divide n = deg f, is fairly well understood, and the upper and lower bounds on the number of decomposable polynomials of degree n match asymptotically. In the wild case, where p does divide n, the bounds are less satisfactory, in particular when p is the smallest prime divisor of n and divides n exactly twice. There is an obvious inclusion-exclusion formula for counting. The main issue is then to determine, under a suitable normalization, the number of collisions, where essentially different components (g, h) yield the same f. In the tame case, Ritt's Second Theorem classifies all collisions of two such pairs. We provide a normal form for collisions of any number of compositions with any number of components. This generalization yields an exact formula for the number of decomposable polynomials of degree n coprime to p. For the wild case, we classify all collisions at degree n = p^2 and obtain the exact number of decomposable polynomials of degree p^2

Counting reducible, powerful, and relatively irreducible multivariate polynomials over finite fields

We present counting methods for some special classes of multivariate polynomials over a finite field, namely the reducible ones, the s-powerful ones (divisible by the s-th power of a nonconstant polynomial), and the relatively irreducible ones (irreducible but reducible over an extension field). One approach employs generating functions, another one uses a combinatorial method. They yield exact formulas and approximations with relative errors that essentially decrease exponentially in the input size.Comment: to appear in SIAM Journal on Discrete Mathematic

Wafer-level uniformity of atomic-layer-deposited niobium nitride thin films for quantum devices

Superconducting niobium nitride thin films are used for a variety of photon detectors, quantum devices, and superconducting electronics. Most of these applications require highly uniform films, for instance, when moving from single-pixel detectors to arrays with a large active area. Plasma-enhanced atomic layer deposition (ALD) of superconducting niobium nitride is a feasible option to produce high-quality, conformal thin films and has been demonstrated as a film deposition method to fabricate superconducting nanowire single-photon detectors before. Here, we explore the property spread of ALD-NbN across a 6-in. wafer area. Over the equivalent area of a 2-in. wafer, we measure a maximum deviation of 1% in critical temperature and 12% in switching current. Toward larger areas, structural characterizations indicate that changes in the crystal structure seem to be the limiting factor rather than film composition or impurities. The results show that ALD is suited to fabricate NbN thin films as a material for large-area detector arrays and for new detector designs and devices requiring uniform superconducting thin films with precise thickness control

Critical temperature modification of low dimensional superconductors by spin doping

Ion implantation of Fe and Mn into Al thin films was used for effective modification of Al superconductive properties. Critical temperature of the transition to superconducting state was found to decrease gradually with implanted Fe concentration. it was found that suppression by Mn implantation much stronger compared to Fe. At low concentrations of implanted ions, suppression of the critical temperature can be described with reasonable accuracy by existing models, while at concentrations above 0.1 at.% a pronounced discrepancy between the models and experiments is observed.Comment: 7 figures, 18 page

CT stress perfusion imaging for detection of haemodynamically relevant coronary stenosis as defined by FFR

Objectives: To evaluate the diagnostic accuracy (DA) of CT-myocardial perfusion imaging (CT-MPI) and a combined approach with CT angiography (CTA) for the detection of haemodynamically relevant coronary stenoses in patients with both suspected and known coronary artery disease. Design: Prospective, non-randomised, diagnostic study. Setting: Academic hospital-based study. Patients: 65 patients (42 men age 70.4 +/- 9) with typical or atypical chest pain. Interventions: CTA and CT-MPI with adenosine stress using a fast dual-source CT system. At subsequent invasive angiography, FFR measurement was performed in coronary arteries to define haemodynamic relevance of stenosis. Main outcome measures: We tried to correlate haemodynamically relevant stenosis (FFR <0.80) to a reduced myocardial blood flow (MBF) as assessed by CT-MPI and determined the DA of CT-MPI for the detection of haemodynamically relevant stenosis. Results: Sensitivity and negative predictive value (NPV) of CTA alone were very high (100% respectively) for ruling out haemodynamically significant stenoses, specificity, Positive predictive value (PPV) and DA were low (43.8, 67.3 and 72%, respectively). CT-MPI showed a significant increase in specificity, PPV and DA for the detection of haemodynamically relevant stenoses (65.6, 74.4 and 81.5%, respectively) with persisting high sensitivity and NPV for ruling out haemodynamically relevant stenoses (97% and 95.5% respectively). The combination of CTA and CT-MPI showed no further increase in detection of haemodynamically significant stenosis compared with CT-MPI alone. Conclusions: Our data suggest that CT-MPI permits the detection of haemodynamically relevant coronary artery stenoses with a moderate DA. CT may, therefore, allow the simultaneous assessment of both coronary morphology and function