Foliations with isolated singularities on Hirzebruch surfaces

We study foliations ℱ {\mathcal{F}} on Hirzebruch surfaces S δ {S_{\delta}} and prove that, similarly to those on the projective plane, any ℱ {\mathcal{F}} can be represented by a bi-homogeneous polynomial affine 1-form. In case ℱ {\mathcal{F}} has isolated singularities, we show that, for δ = 1 {\delta=1} , the singular scheme of ℱ {\mathcal{F}} does determine the foliation, with some exceptions that we describe, as is the case of foliations in the projective plane. For δ ≠ 1 {\delta\neq 1} , we prove that the singular scheme of ℱ {\mathcal{F}} does not determine the foliation. However, we prove that, in most cases, two foliations ℱ {\mathcal{F}} and ℱ ′ {\mathcal{F}^{\prime}} given by sections s and s ′ {s^{\prime}} have the same singular scheme if and only if s ′ = Φ ⁢ ( s ) {s^{\prime}=\Phi(s)} , for some global endomorphism Φ of the tangent bundle of S δ {S_{\delta}} .

Universality of High-Strength Tensors

A theorem due to Kazhdan and Ziegler implies that, by substituting linear forms for its variables, a homogeneous polynomial of sufficiently high strength specialises to any given polynomial of the same degree in a bounded number of variables. Using entirely different techniques, we extend this theorem to arbitrary polynomial functors. As a corollary of our work, we show that specialisation induces a quasi-order on elements in polynomial functors, and that among the elements with a dense orbit there are unique smallest and largest equivalence classes in this quasi-order.

Fiscal reaction functions for the advanced economies revisited

We revisit the relationship between the primary balances/GDP and debt/GDP ratios (fiscal reaction function, FRF), in the advanced economies, showing that using adequate tests and estimators leads to question the validity of the current consensus. Using data for 1961–2019, we find that long-run FRFs exist only in a small number of advanced economies (Belgium, Germany, Greece, Norway, Portugal and Sweden), with polynomial effects with fiscal fatigue only in Germany. These results warn against the widespread practice of estimating homogeneous polynomial panel FRFs. Limiting the sample to 1961–2007, thus excluding the 2008 crisis and its aftermath, FRFs hold also in Canada, Ireland, Italy (polynomial), Spain and USA, though not in Germany, and the coefficients are generally larger. Particularly, after 2008 European Union countries appear somehow to have been more likely to implement FRFs.

Two methods for the maximization of homogeneous polynomials over the simplex

The paper deals with the numerical solution of the problem P to maximize a homogeneous polynomial over the unit simplex. We discuss the convergence properties of the so-called replicator dynamics for solving P. We further examine an ascent method, which also makes use of the replicator transformation. Numerical experiments with polynomials of different degrees illustrate the theoretical convergence results.