Development of Land Markets in Selected EU-Countries and Land Ownership Strategies on the Farm Level

The development of the land markets for selected European Countries (B, DK, D-W, F, NL) and the effects of agricultural policy reforms on land prices are analysed in "Part I - Land Markets at the Country Level" of this article. The variables describing the agricultural land markets are land prices, prices for rental land and the share of rented land. Over a longer period of about two decades, the prices for agricultural land have been decreasing in real terms in general (exception NL) while rental prices have been more stable. A regression analysis shows increasing price effects on prices for agricultural land and on rental prices due to the ha-premiums, which were introduced by the Common Agricultural Policy(CAP)-Reforms in 1992. In "Part II - Farm Level Strategies for Land Purchase" the economic effects of changing land ownership are analysed using the data base of Part I. Recommendations for farmers are to find out if long term rental contracts are available to avoid tying up capital. If land market prices are lower than a certain threshold, land can be bought considering the financial liquidity of the enterprise. Selling land could be appropriate to avoid financial liquidity stress, but this should be only temporary and restricted to a marginal amount of land stocks. If profitability of crop production is relatively low or land prices are relative high and rental contracts terminated, an entrepreneur should seek to rent new land, if possible, or find alternative investments which yield higher profits than crop production.Land Economics/Use,

Commutative algebraic groups and pp-adic linear forms

Let $G$ be a commutative algebraic group defined over a number field $K$ that is disjoint over $K$ to $\mathbb G_a$ and satisfies the condition of semistability. Consider a linear form $l$ on the Lie algebra of $G$ with algebraic coefficients and an algebraic point $u$ in a $p$-adic neighbourhood of the origin with the condition that $l$ does not vanish at $u$. We give a lower bound for the $p$-adic absolute value of $l(u)$ which depends up to an effectively computable constant only on the height of the linear form, the height of the point $u$ and $p$.Comment: This is a preprint of the Materials accepted for publication in "Acta Arithmetica

Decomposable polynomials in second order linear recurrence sequences

We study elements of second order linear recurrence sequences $(G_n)_{n= 0}^{\infty}$ of polynomials in $\mathbb{C}[x]$ which are decomposable, i.e. representable as $G_n=g\circ h$ for some $g, h\in \mathbb{C}[x]$ satisfying $\operatorname{deg}g,\operatorname{deg}h>1$. Under certain assumptions, and provided that $h$ is not of particular type, we show that $\operatorname{deg}g$ may be bounded by a constant independent of $n$, depending only on the sequence.Comment: 26 page

Diophantine triples in linear recurrence sequences of Pisot type

The study of Diophantine triples taking values in linear recurrence sequences is a variant of a problem going back to Diophantus of Alexandria which has been studied quite a lot in the past. The main questions are, as usual, about existence or finiteness of Diophantine triples in such sequences. Whilst the case of binary recurrence sequences is almost completely solved, not much was known about recurrence sequences of larger order, except for very specialized generalizations of the Fibonacci sequence. Now, we will prove that any linear recurrence sequence with the Pisot property contains only finitely many Diophantine triples, whenever the order is large and a few more not very restrictive conditions are met.Comment: 25 pages. arXiv admin note: text overlap with arXiv:1602.0823

On a parametric family of Thue inequalities over function fields

In this paper we completely solve a family of Thue inequalities defined over the field of functions $\mathbb{C}(T)$, namely deg (X4−4cX3Y+(6c+2)X2Y2+4cXY3+Y4) ≤ deg c, where the solutions x,y come from the ring $\mathbb{C}\left[ T\right]$ and the parameter $c\in \mathbb{C}\left[ T\right]$ is some non-constant polynomia

Integral zeros of a polynomial with linear recurrences as coefficients

Let $K$ be a number field, $S$ a finite set of places of $K$, and $\mathcal{O}_S$ be the ring of $S$-integers. Moreover, let $$G_n^{(0)} Z^d + \cdots + G_n^{(d-1)} Z + G_n^{(d)}$$ be a polynomial in $Z$ having simple linear recurrences of integers evaluated at $n$ as coefficients. Assuming some technical conditions we give a description of the zeros $(n,z) \in \mathbb{N} \times \mathcal{O}_S$ of the above polynomial. We also give a result in the spirit of Hilbert irreducibility for such polynomials.Comment: 13 page

A function field variant of Pillai's problem

In this paper, we consider a variant of Pillai's problem over function fields $F$ in one variable over $\mathbb{C}$. For given simple linear recurrence sequences $G_n$ and $H_m$, defined over $F$ and satisfying some weak conditions, we will prove that the equation $G_n - H_m = f$ has only finitely many solutions $(n,m) \in \mathbb{N}^2$ for any non-zero $f \in F$, which can be effectively bounded. Furthermore, we prove that under suitable assumptions there are only finitely many effectively computable $f$ with more than one representation of the form $G_n - H_m$.Comment: 13 page

Yet another SS-unit variant of Diophantine tuples

We show that there are only finitely many triples of integers $0 < a < b < c$ such that the product of any two of them is the value of a given polynomial with integer coefficients evaluated at an $S$-unit that is also a positive integer. The proof is based on a result of Corvaja and Zannier and thus is ultimately a consequence of the Schmidt subspace theorem.Comment: 9 page