Showing papers on "Ring of integers published in 2018"
TL;DR: In this article, a new cohomology theory for proper smooth (formal) schemes over the ring of integers was proposed, which takes values in a mixed-characteristic analogue of Dieudonne modules, previously defined by Fargues as a version of Breuil-Kisin modules.
Abstract: We construct a new cohomology theory for proper smooth (formal) schemes over the ring of integers of $\mathbf {C}_{p}$
. It takes values in a mixed-characteristic analogue of Dieudonne modules, which was previously defined by Fargues as a version of Breuil–Kisin modules. Notably, this cohomology theory specializes to all other known $p$
-adic cohomology theories, such as crystalline, de Rham and etale cohomology, which allows us to prove strong integral comparison theorems. The construction of the cohomology theory relies on Faltings’ almost purity theorem, along with a certain functor $L\eta $
on the derived category, defined previously by Berthelot–Ogus. On affine pieces, our cohomology theory admits a relation to the theory of de Rham–Witt complexes of Langer–Zink, and can be computed as a $q$
-deformation of de Rham cohomology.
145 citations
29 Apr 2018
TL;DR: It is proved that the (decision/search) dual to primal reduction from Lyubashevsky et al. can be implemented with a small error rate growth for all rings and the resulting reduction is non-uniform polynomial time.
Abstract: The Ring Learning With Errors problem (\(\mathsf {RLWE}\)) comes in various forms. Vanilla \(\mathsf {RLWE}\) is the decision dual-\(\mathsf {RLWE}\) variant, consisting in distinguishing from uniform a distribution depending on a secret belonging to the dual \(\mathcal {O}_K^{\vee }\) of the ring of integers \(\mathcal {O}_K\) of a specified number field K. In primal-\(\mathsf {RLWE}\), the secret instead belongs to \(\mathcal {O}_K\). Both decision dual-\(\mathsf {RLWE}\) and primal-\(\mathsf {RLWE}\) enjoy search counterparts. Also widely used is (search/decision) Polynomial Learning With Errors (\(\mathsf {PLWE}\)), which is not defined using a ring of integers \(\mathcal {O}_K\) of a number field K but a polynomial ring \(\mathbb {Z}[x]/f\) for a monic irreducible \(f \in \mathbb {Z}[x]\). We show that there exist reductions between all of these six problems that incur limited parameter losses. More precisely: we prove that the (decision/search) dual to primal reduction from Lyubashevsky et al. [EUROCRYPT 2010] and Peikert [SCN 2016] can be implemented with a small error rate growth for all rings (the resulting reduction is non-uniform polynomial time); we extend it to polynomial-time reductions between (decision/search) primal \(\mathsf {RLWE}\) and \(\mathsf {PLWE}\) that work for a family of polynomials f that is exponentially large as a function of \(\deg f\) (the resulting reduction is also non-uniform polynomial time); and we exploit the recent technique from Peikert et al. [STOC 2017] to obtain a search to decision reduction for \(\mathsf {RLWE}\) for arbitrary number fields. The reductions incur error rate increases that depend on intrinsic quantities related to K and f.
53 citations
TL;DR: In this article, the authors studied the curvature of the Schmidt arrangement in the Bianchi group and showed that it is a disjoint union of all primitive integral groups of Apollonian packings.
Abstract: We study the orbit of $\widehat{\mathbb{R}}$ under the Mobius action of the Bianchi group $\operatorname{PSL}_2(\mathcal{O}_K)$ on $\widehat{\mathbb{C}}$, where $\mathcal{O}_K$ is the ring of integers of an imaginary quadratic field $K$. The orbit $\mathcal{S}_K$, called a Schmidt arrangement, is a geometric realisation, as an intricate circle packing, of the arithmetic of $K$. We give a simple geometric characterisation of certain subsets of $\mathcal{S}_K$ generalizing Apollonian circle packings, and show that $\mathcal{S}_K$, considered with orientations, is a disjoint union of all primitive integral such $K$-Apollonian packings. These packings are described by a new class of thin groups of arithmetic interest called $K$-Apollonian groups. We make a conjecture on the curvatures of these packings, generalizing the local-to-global conjecture for Apollonian circle packings.
27 citations
16 Feb 2018
TL;DR: In this article, it was shown that the Galois group of polynomials is large enough (in a precise sense) as a function of n to be irreducible.
Abstract: Let $K$ be a number field with ring of integers $\mathcal O_K$, and let $\{f_k\}_{k\in \mathbb N}\subseteq \mathcal O_K[x]$ be a sequence of monic polynomials such that for every $n\in \mathbb N$, the composition $f^{(n)}=f_1\circ f_2\circ\ldots\circ f_n$ is irreducible. In this paper we show that if the size of the Galois group of $f^{(n)}$ is large enough (in a precise sense) as a function of $n$, then the set of primes $\mathfrak p\subseteq\mathcal O_K$ such that every $f^{(n)}$ is irreducible modulo $\mathfrak p$ has density zero. Moreover, we prove that the subset of polynomial sequences such that the Galois group of $f^{(n)}$ is large enough has density 1, in an appropriate sense, within the set of all polynomial sequences.
22 citations
18 Jun 2018
TL;DR: Construction A of lattices is extended to the ring of algebraic integers of a general imaginary quadratic field that may not form a principal ideal domain (PID) and it is shown that such a construction can produce good lattices for coding in the sense of Poltyrev and for MSE quantization.
Abstract: In this paper, we extend Construction A of lattices to the ring of algebraic integers of a general imaginary quadratic field that may not form a principal ideal domain (PID). We show that such a construction can produce good lattices for coding in the sense of Poltyrev and for MSE quantization. As an application, we then apply the proposed lattices to the compute-and-forward paradigm with limited feedback. Without feedback, compute-and-forward is typically realized with lattice codes over the ring of integers, the ring of Gaussian integers, or the ring of Eisenstein integers, which are all PIDs. A novel scheme called adaptive compute-and-forward is proposed to exploit the limited feedback about the channel state by working with the best ring of imaginary quadratic integers. Simulation results show that by adaptively choosing the best ring among the considered ones according to the limited feedback, the proposed adaptive compute-and-forward provides a better performance than that provided by the conventional compute-and-forward scheme which works over Gaussian or Eisenstein integers solely.
16 citations
31 Jan 2018
TL;DR: In this article, it was shown that there is a linear bound for the largest p-power order of a torsion point defined over a number field of degree d, which in fact is linear in the maximum ramification index of a prime ideal of the ring of integers F over (p).
Abstract: Let $$d\ge 1$$
be fixed. Let F be a number field of degree d, and let E / F be an elliptic curve. Let $$E(F)_{\text {tors}}$$
be the torsion subgroup of E(F). In 1996, Merel proved the uniform boundedness conjecture, i.e., there is a constant B(d), which depends on d but not on the chosen field F or on the curve E / F, such that the size of $$E(F)_{\text {tors}}$$
is bounded by B(d). Moreover, Merel gave a bound (exponential in d) for the largest prime that may be a divisor of the order of $$E(F)_{\text {tors}}$$
. In 1996, Parent proved a bound (also exponential in d) for the largest p-power order of a torsion point that may appear in $$E(F)_{\text {tors}}$$
. It has been conjectured, however, that there is a bound for the size of $$E(F)_{\text {tors}}$$
that is polynomial in d. In this article we show that under certain hypotheses there is a linear bound for the largest p-power order of a torsion point defined over F, which in fact is linear in the maximum ramification index of a prime ideal of the ring of integers F over (p).
15 citations
TL;DR: In this article, a cohomological construction of the Breuil-Kisin modules attached to the $p$-adic etale cohomology was proposed for a perfect field of characteristic $p>0.
Abstract: For a perfect field $k$ of characteristic $p>0$ and a smooth and proper formal scheme $\mathscr{X}$ over the ring of integers of a finite and totally ramified extension $K$ of $W(k)[1/p]$, we propose a cohomological construction of the Breuil-Kisin modules attached to the $p$-adic etale cohomology $H^i_{\mathrm{et}}(\mathscr{X}_{\overline{K}},\mathbf{Z}_p)$. We then prove that our proposal works when $p>2$, $i < p-1$, and the crystalline cohomology of the special fiber of $\mathscr{X}$ is torsion-free in degrees $i$ and $i+1$.
15 citations
TL;DR: In this paper, the authors extend the definition of Int K -decomposability to D-algebras and show that a D-module basis for A is also an Int K-decompositional basis for Int K (A ) if A is isomorphic to Int K(A ) ⊗ D A.
13 citations
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TL;DR: In this article, a toroidal embedding associated to the Newton polygon of a smooth projective curve over a discretely valued field K is constructed for the case of affine equations f(x,y) = 0.
Abstract: Let C be a smooth projective curve over a discretely valued field K, defined by an affine equation f(x,y)=0. We construct a model of C over the ring of integers of K using a toroidal embedding associated to the Newton polygon of f. We show that under 'generic' conditions it is regular with normal crossings, and determine when it is minimal, the global sections of its relative dualising sheaf, and the tame part of the first etale cohomology of C.
12 citations
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TL;DR: In this article, it was shown that the Diophantine problem in the ring of algebraic integers is polynomial-time reducible to the problem in finite systems of equations (e-interpretable).
Abstract: We study the Diophantine problem (decidability of finite systems of equations) in different classes of finitely generated solvable groups (nilpotent, polycyclic, metabelian, free solvable, etc), which satisfy some natural "non-commutativity" conditions. For each group $G$ in one of these classes, we prove that there exists a ring of algebraic integers $O$ that is interpretable in $G$ by finite systems of equations (e-interpretable), and hence that the Diophantine problem in $O$ is polynomial time reducible to the Diophantine problem in $G$. One of the major open conjectures in number theory states that the Diophantine problem in any such $O$ is undecidable. If true this would imply that the Diophantine problem in any such $G$ is also undecidable. Furthermore, we show that for many particular groups $G$ as above, the ring $O$ is isomorphic to the ring of integers $\mathbb{Z}$, so the Diophantine problem in $G$ is, indeed, undecidable. This holds, in particular, for free nilpotent or free solvable non-abelian groups, as well as for non-abelian generalized Heisenberg groups and uni-triangular groups $UT(n,\mathbb{Z}), n \geq 3$. Then we apply these results to non-solvable groups that contain non-virtually abelian maximal finitely generated nilpotent subgroups. For instance, we show that the Diophantine problem is undecidable in the groups $GL(3,\mathbb{Z}), SL(3,\mathbb{Z}), T(3,\mathbb{Z})$.
11 citations
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TL;DR: In this article, the authors provide techniques to calculate with non-commutative polynomials in the free field of rational functions, where the number of irreducible factors is fixed.
Abstract: Long before we learn to construct the field of rational numbers (out of the ring of integers) at university, we learn how to calculate with fractions at school. When it comes to "numbers", we are used to a commutative multiplication, for example 2*3=6=3*2. On the other hand --even before we can write-- we learn to talk (in a language) using words, consisting of purely non-commuting "letters" (or symbols), for example "xy" is not equal to "yx" (with the concatenation as multiplication). Now, if we combine numbers (from a field) with words (from the free monoid of an alphabet) we get non-commutative polynomials which form a ring (with "natural" addition and multiplication), namely the free associative algebra. Adding or multiplying polynomials is easy, for example (2/3*xy+z)+1/3*xy=xy+z or 2*x(yx+3*z)=2*xyx+6*xz. Although the integers and the non-commutative polynomials look rather different, they share many properties, for example the unique number of irreducible factors: x(1-yx)=x-xyx=(1-xy)x. However, the construction of the universal field of fractions (aka "free field") of the free associative algebra is highly non-trivial (but really beautiful). Therefore we provide techniques (building on the work of Cohn and Reutenauer) to calculate with free fractions (representing elements in the free field or "skew field of non-commutative rational functions") to be able to explore a fascinating non-commutative world.
TL;DR: In this article, it was shown that the growth of the smallest integer such that an integral quadratic form can be written as a sum of squares of integral linear forms is at most an exponential in the size of the integral linear form.
Abstract: For each positive integer $n$, let $g_{\mathbb Z}(n)$ be the smallest integer such that if an integral quadratic form in $n$ variables can be written as a sum of squares of integral linear forms, then it can be written as a sum of $g_{\mathbb Z}(n)$ squares of integral linear forms. We show that as $n$ goes to infinity, the growth of $g_{\mathbb Z}(n)$ is at most an exponential of $\sqrt{n}$. Our result improves the best known upper bound on $g_{\mathbb Z}(n)$ which is in the order of an exponential of $n$. We also define an analogous number $g_{\mathcal O}^*(n)$ for writing hermitian forms over the ring of integers $\mathcal O$ of an imaginary quadratic field as sums of norms of integral linear forms, and when the class number of the imaginary quadratic field is 1, we show that the growth of $g_{\mathcal O}^*(n)$ is at most an exponential of $\sqrt{n}$. We also improve results of Conway-Sloane and Kim-Oh on $s$-integral lattices.
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TL;DR: In this paper, it was shown that the first order theory with constants of all the aforementioned structures $R$ is undecidable, and in passing it was conjectured that it is so.
Abstract: We study systems of equations in different families of rings and algebras. In each such structure $R$ we interpret by systems of equations (e-interpret) a ring of integers $O$ of a global field. The long standing conjecture that $\mathbb{Z}$ is always e-interpretable in $O$ then carries over to $R$, and if true it implies that the Diophantine problem in $R$ is undecidable. The conjecture is known to be true if $O$ has positive characteristic, i.e. if $O$ is not a ring of algebraic integers. As a corollary we describe families of structures where the Diophantine problem is undecidable, and in other cases we conjecture that it is so. In passing we obtain that the first order theory with constants of all the aforementioned structures $R$ is undecidable.
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TL;DR: In this article, the authors study K3 surfaces with complex multiplication and provide an algorithm that given a number field and a CM number field, returns a finite lists of groups which contain the moduli of any K3 surface with CM by the ring of integers of the number field.
Abstract: We study K3 surfaces with complex multiplication following the classical work of Shimura on CM abelian varieties. After we translate the problem in terms of the arithmetic of the CM field and its ideles, we proceed to study some abelian extensions that arise naturally in this context. We then make use of our computations to determine the fields of moduli of K3 surfaces with CM and to classify their Brauer groups. More specifically, we provide an algorithm that given a number field $K$ and a CM number field $E$, returns a finite lists of groups which contains $\mathrm{Br}(\overline{X})^{G_K}$ for any K3 surface $X/K$ that has CM by the ring of integers of $E$. We run our algorithm when $E$ is a quadratic imaginary field (a condition that translates into $X$ having maximal Picard rank) generalizing similar computations already appearing in the literature.
TL;DR: A class of integer codes capable of correcting burst and random asymmetric (1 → 0) errors within a b-bit byte is presented, defined over the ring of integers modulo 2b −1.
Abstract: In most communication networks, error probabilities 1 → 0 and 0 → 1 are equally likely to occur. However, in some optical networks, such as local and access networks, this is not the case. In these networks, the number of received photons never exceeds the number of transmitted ones. Hence, if the receiver operates correctly, only 1 → 0 errors can occur. Motivated by this fact, in this paper, we present a class of integer codes capable of correcting burst and random asymmetric (1 → 0) errors within a b-bit byte. Unlike classical codes, the proposed codes are defined over the ring of integers modulo 2b −1. As a result, they have the potential to be implemented in software without any hardware assist.
TL;DR: The aim of this paper is to generalize the well-known Halton sequences from integer bases to rational number bases and to translate this concept of Halton-type sequences in rational bases from the ring of integers to theRing of polynomials over a finite field.
TL;DR: In this paper, a complete proof of a duality theorem for the fppf cohomology of either a curve over a finite field or a ring of integers of a number field is given.
Abstract: We provide a complete proof of a duality theorem for the fppf cohomology of either a curve over a finite field or a ring of integers of a number field, which extends the classical Artin-Verdier Theorem in etale cohomology. We also prove some finiteness and vanishing statements.
TL;DR: In this article, it was shown that for any rational prime q and any positive rational integer m, algebraic integers are definable in any Galois extension of Q where the degree of any finite subextension is not divisible by qm.
Abstract: We extend results of Videla and Fukuzaki to define algebraic integers in large classes of infinite algebraic extensions of Q and use these definitions for some of the fields to show the first-order undecidabilitv. We also obtain a structural sufficient condition for definability of the ring of integers over its field of fractions. In particular, we show that the following propositions hold: (1) For any rational prime q and any positive rational integer m. algebraic integers are definable in any Galois extension of Q where the degree of any finite subextension is not divisible by qm. (2) Given a prime q, and an integer m > 0, algebraic integers are definable in a cyclotomic extension (and any of its subfields) generated by any set $$\{ {\zeta _{{p^l}}}|l \in {Z_{ > 0,}}P
e q$$
is any prime such that qm +1 ∧(p — 1)}. (3) The first-order theory of Any Abelina Extension of Q With Finitely Many Rational Primes is undecidable and rational integers are definable in these extensions. We also show that under a condition on the splitting of one rational Q generated elliptic curve over the field in question is enough to have a definition of Z and to show that the field is undecidable.
TL;DR: In this article, the authors prove automorphy lifting for geometric representations of elliptic curves, such that the residual representation is totally odd and induced from a character of the absolute Galois group of the quadratic subfield $K$ of $F(\zeta_p)/F$ such that such representations fail the Taylor-Wiles hypothesis.
Abstract: We prove automorphy lifting results for geometric representations $\rho:G_F \rightarrow GL_2(\mathcal{O})$, with $F$ a totally real field, and $\mathcal{O}$ the ring of integers of a finite extension of $\mathbb{Q}_p$ with $p$ an odd prime, such that the residual representation $\bar{\rho}$ is totally odd and induced from a character of the absolute Galois group of the quadratic subfield $K$ of $F(\zeta_p)/F$. Such representations fail the Taylor-Wiles hypothesis and the patching techniques to prove automorphy do not work. We apply this to automorphy of elliptic curves $E$ over $F$, when $E$ has no $F$ rational 7-isogeny and such that the image of $G_F$ acting on $E[7]$ normalizes a split Cartan subgroup of $GL_2(\mathbb{F}_7)$.
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TL;DR: In this article, a cohomology theory on separated quasi-compact smooth rigid spaces over the ring of integers of a finite extension of the Fargues-Fontaine curve is presented.
Abstract: We explain how to construct a cohomology theory on the category of separated quasi-compact smooth rigid spaces over $\mathbf{C}_p$ (or more general base fields), taking values in the category of vector bundles on the Fargues-Fontaine curve, which extends (in a suitable sense) Hyodo-Kato cohomology when the rigid space has a semi-stable proper formal model over the ring of integers of a finite extension of $\mathbf{Q}_p$. This cohomology theory factors through the category of rigid analytic motives of Ayoub.
TL;DR: In this article, an efficient method to reduce the resolution of the relative Thue inequalities to the resolution for absolute Thue inequality of type |F(x, y)|≤ k (x and y ∈ Z¬_M] was proposed.
Abstract: Let F(x, y) be an irreducible binary form of degree ≥ 3 with integer coefficients and with real roots. Let M be an imaginary quadratic field, with ring of integers Z_M. Let K > 0. We describe an efficient method how to reduce the resolution of the relative Thue inequalities |F(x, y)|≤ K (x, y ∈Z¬_M) to the resolution of absolute Thue inequalities of type |F(x, y)|≤ k (x, y ∈Z). We illustrate our method with an explicit example.
TL;DR: In this paper, it was shown that the abscissas of convergence of the representation zeta functions of the special linear groups over the ring of integers are bounded above by 2.
Abstract: We prove that the numbers of irreducible n-dimensional complex continuous representations of the special linear groups over p-adic integers grow slower than the square of n. We deduce that the abscissas of convergence of the representation zeta functions of the special linear groups over the ring of integers are bounded above by 2. In order to show these results we prove also that if G is a connected, simply connected, semi-simple algebraic group defined over the field of rational numbers, then the G-representation variety of the fundamental group of a compact Riemann surface of genus n has rational singularities if and only if the G-character variety has rational singularities.
TL;DR: In this paper, the existence of a large explicit infinite family of quadratic twists of the Gross elliptic curve whose complex series does not vanish at the s = 1 constant is proved.
Abstract: Let $q$ be any prime $\equiv 7 \mod 16$, $K = \mathbb{Q}(\sqrt{-q})$, and let $H$ be the Hilbert class field of $K$. Let $A/H$ be the Gross elliptic curve defined over $H$ with complex multiplication by the ring of integers of $K$. We prove the existence of a large explicit infinite family of quadratic twists of $A$ whose complex $L$-series does not vanish at $s=1$. This non-vanishing theorem is completely new when $q > 7$. Its proof depends crucially on the results established in our earlier paper for the Iwasawa theory at the prime $p=2$ of the abelian variety $B/K$, which is the restriction of scalars from $H$ to $K$ of the elliptic curve $A$.
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TL;DR: In this paper, it was shown that the Satake category of G can be described via Iwahori-Whittaker perverse sheaves on the affine Grassmannian.
Abstract: In this paper we prove, for G a connected reductive algebraic group satisfying a technical assumption, that the Satake category of G (with coefficients in a finite field, a finite extension of Q_l, or the ring of integers of such a field) can be described via Iwahori-Whittaker perverse sheaves on the affine Grassmannian. As an application, we confirm a conjecture of Juteau-Mautner-Williamson describing the tilting objects in the Satake category.
TL;DR: In this article, Bergelson and Moreira developed a new approach based on ultrafilter limits, which allows one to refine and extend the results obtained in Bergelson-Moreira, op. cit., to a more general situation involving measure preserving actions of the non-amenable affine semigroups of a large class of integral domains.
Abstract: Ergodic and combinatorial results obtained in Bergelson and Moreira [Ergodic theorem involving additive and multiplicative groups of a field and patterns. Ergod. Th. & Dynam. Sys. to appear, published online 6 October 2015, doi:10.1017/etds.2015.68], involved measure preserving actions of the affine group of a countable field . In this paper, we develop a new approach, based on ultrafilter limits, which allows one to refine and extend the results obtained in Bergelson and Moreira, op. cit., to a more general situation involving measure preserving actions of the non-amenable affine semigroups of a large class of integral domains. (The results and methods in Bergelson and Moreira, op. cit., heavily depend on the amenability of the affine group of a field.) Among other things, we obtain, as a corollary of an ultrafilter ergodic theorem, the following result. Let be a number field and let be the ring of integers of . For any finite partition , there exists such that, for many and many , .
13 Jun 2018
TL;DR: Let an algebraic lattice coming from a projective module over the ring of integers of a number field, known as the zero locus of polynula, be the zeroLocator of a finite-of-polynula content-type.
Abstract: Let $\Lambda \subset \mathbb R^n$ be an algebraic lattice, coming from a projective module over the ring of integers of a number field $K$. Let $\mathcal Z \subset \mathbb R^n$ be the zero locus of a finite collection of polynomials such that $\Lambda
subseteq \mathcal Z$ or a finite union of proper full-rank sublattices of $\Lambda$. Let $K_1$ be the number field generated over $K$ by coordinates of vectors in $\Lambda$, and let $L_1,\dots,L_t$ be linear forms in $n$ variables with algebraic coefficients satisfying an appropriate linear independence condition over $K_1$. For each $\varepsilon > 0$ and $\boldsymbol a \in \mathbb R^n$, we prove the existence of a vector $\boldsymbol x \in \Lambda \setminus \mathcal Z$ of explicitly bounded sup-norm such that $$\| L_i(\boldsymbol x) - a_i \| < \varepsilon$$ for each $1 \leq i \leq t$, where $\|\ \|$ stands for the distance to the nearest integer. The bound on sup-norm of $\boldsymbol x$ depends on $\varepsilon$, as well as on $\Lambda$, $K$, $\mathcal Z$ and heights of linear forms. This presents a generalization of Kronecker's approximation theorem, establishing an effective result on density of the image of $\Lambda \setminus \mathcal Z$ under the linear forms $L_1,\dots,L_t$ in the $t$-torus~$\mathbb R^t/\mathbb Z^t$.
TL;DR: In this paper, the authors generalize the definition and properties of root systems to complex reflection groups, where roots become rank one projective modules over the ring of integers of a number field.
TL;DR: In this paper, the authors studied integer solutions of an implicit linear inhomogeneous first order difference equation bxn+1 = axn + fn, and showed that for a = 1, a typical (in the natural topological sense) equation has no integer solutions.
Abstract: We study solutions in integers of an implicit linear inhomogeneous first order difference equation bxn+1 = axn + fn. Based on the p-adic topology on the ring of integers, we obtain a criterion for the existence of solutions and show that for a = 1 a typical (in the natural topological sense) equation has no integer solutions.
TL;DR: In this paper, the Montes algorithm was used to show that a root of polynomials (i.e., a polynomial in the families f(a,b) and g(c,d) can be used as a generator for a power integral basis of the ring of integers.
Abstract: Consider the integral polynomials $f_{a,b}(x)=x^4+ax+b$ and $g_{c,d}(x)=x^4+cx^3+d$. Suppose $f_{a,b}(x)$ and $g_{c,d}(x)$ are irreducible, $b\mid a$, and the integers $b$, $d$, $256d-27c^4$, and $\dfrac{256b^3-27a^4}{\gcd(256b^3,27a^4)}$ are all square-free. Using the Montes algorithm, we show that a root of $f_{a,b}(x)$ or $g_{c,d}(x)$ defines a monogenic extension of $\mathbb{Q}$ and serves as a generator for a power integral basis of the ring of integers. In fact, we show monogeneity for slightly more general families. Further, we obtain lower bounds on the density of polynomials generating monogenic $S_4$ fields within the families $f_{b,b}(x)$ and $g_{1,d}(x)$.
TL;DR: In this paper, it was shown that the ring of integers $\mathbb{Z}$ is interpretable by positive existential formulas in such free Lie algebras over a field of characteristic zero.
Abstract: In this paper we prove undecidability of finite systems of equations in free Lie algebras of rank at least three over an arbitrary field. We show that the ring of integers $\mathbb{Z}$ is interpretable by positive existential formulas in such free Lie algebras over a field of characteristic zero.