Duality of weighted anisotropic Besov and Triebel–Lizorkin spaces

Let A be an expansive dilation on ${{\mathbb R}^n}$ and w a Muckenhoupt ${\mathcal A_\infty(A)}$ weight. In this paper, for all parameters ${\alpha\in{\mathbb R} }$ and ${p,q\in(0,\infty)}$ , the authors identify the dual spaces of weighted anisotropic Besov spaces ${\dot B^\alpha_{p,q}(A;w)}$ and Triebel?CLizorkin spaces ${\dot F^\alpha_{p,q}(A;w)}$ with some new weighted Besov-type and Triebel?CLizorkin-type spaces. The corresponding results on anisotropic Besov spaces ${\dot B^\alpha_{p,q}(A; \mu)}$ and Triebel?CLizorkin spaces ${\dot F^\alpha_{p,q}(A; \mu)}$ associated with ${\rho_A}$ -doubling measure??? are also established. All results are new even for the classical weighted Besov and Triebel?CLizorkin spaces in the isotropic setting. In particular, the authors also obtain the ${\varphi}$ -transform characterization of the dual spaces of the classical weighted Hardy spaces on ${{\mathbb R}^n}$ .     

Extinction profile of the logarithmic diffusion equation

Let ${N \geq 3}$ and u be the solution of u _t = Δ log u in ${\mathbb{R}^N \times (0, T)}$ with initial value u ₀ satisfying ${B_{k_1}(x, 0) \leq u_{0} \leq B_{k_2}(x, 0)}$ for some constants k ₁ > k ₂ > 0 where ${B_k(x, t) = 2(N - 2)(T - t)_{+}^{N/(N - 2)}/(k + (T - t)_{+}^{2/(N - 2)}|x|^{2})}$ is the Barenblatt solution for the equation and ${u_0 - B_{k_0} \in L^{1}(\mathbb{R}^{N})}$ for some constant k ₀ > 0 if ${N \geq 4}$ . We give a new different proof on the uniform convergence and ${L^1(\mathbb{R}^N)}$ convergence of the rescaled function ${\tilde{u}(x, s) = (T - t)^{-N/(N - 2)}u(x/(T - t)^{-1/(N - 2)}, t), s = -{\rm log}(T - t)}$ , on ${\mathbb{R}^N}$ to the rescaled Barenblatt solution ${\tilde{B}_{k_0}(x) = 2(N - 2)/(k_0 + |x|^{2})}$ for some k ₀ > 0 as ${s \rightarrow \infty}$ . When ${N \geq 4, 0 \leq u_0(x) \leq B_{k_0}(x, 0)}$ in ${\mathbb{R}^N}$ , and ${|u_0(x) - B_{k_0}(x, 0)| \leq f \in L^{1}(\mathbb{R}^{N})}$ for some constant k ₀ > 0 and some radially symmetric function f, we also prove uniform convergence and convergence in some weighted L ¹ space in ${\mathbb{R}^N}$ of the rescaled solution ${\tilde{u}(x, s)}$ to ${\tilde{B}_{k_0}(x)}$ as ${s \rightarrow \infty}$ .     

The Hopf–Lax formula in Carnot groups: a control theoretic approach

The purpose of this paper is to bring a new light on the state-dependent Hamilton–Jacobi equation and its connection with the Hopf–Lax formula in the framework of a Carnot group $(\mathbf G ,\circ ).$ The equation we shall consider is of the form $$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} u_{t}+ \Psi (X_{1}u, \ldots , X_{m}u)=0\qquad &{}(x,t)\in \mathbf G \times (0,\infty ) \\ {u}(x,0)=g(x)&{}x\in \mathbf G , \end{array} \right. \end{aligned}$$ where $X_{1},\ldots , X_{m}$ are a basis of the first layer of the Lie algebra of the group $\mathbf G ,$ and $\Psi : \mathbb{R }^{m} \rightarrow \mathbb{R }$ is a superlinear, convex function. The main result shows that the unique viscosity solution of the Hamilton–Jacobi equation can be given by the Hopf–Lax formula $$\begin{aligned} u(x,t) = \inf _{y\in \mathbf G }\left\{ t \Psi ^\mathbf{G }\left( \delta _{\frac{1}{t}}(y^{-1}\circ x)\right) + g(y) \right\} , \end{aligned}$$ where $\Psi ^\mathbf{G }:\mathbf G \rightarrow \mathbb{R }$ is the $\mathbf G $ -Legendre–Fenchel transform of $\Psi ,$ defined by a control theoretical approach. We recover, as special cases, some known results like the classical Hopf–Lax formula in the Euclidean spaces $\mathbb{R }^n,$ showing that $\Psi ^{\mathbb{R }^n}$ is the Legendre–Fenchel transform $\Psi ^*$ of $\Psi ;$ moreover, we recover the result by Manfredi and Stroffolini in the Heisenberg group for particular Hamiltonian function $\Psi .$ In this paper we follow an optimal control problem approach and we obtain several properties for the value functions $u$ and $\Psi ^\mathbf G .$      

Growth estimates for orbits of self-adjoint groups

Let $G$ denote a closed, connected, self-adjoint, noncompact subgroup of $GL(n,\mathbb R )$ , and let $d_{R}$ and $d_{L}$ denote respectively the right and left invariant Riemannian metrics defined by the canonical inner product on $M(n,\mathbb R ) = T_{I} GL(n,\mathbb R )$ . Let $v$ be a nonzero vector of $\mathbb R ^{n}$ such that the orbit $G(v)$ is unbounded in $\mathbb R ^{n}$ . Then the function $g \rightarrow d_{R}(g, G_{v})$ is unbounded, where $G_{v} = \{g \in G : g(v) = v \}$ , and we obtain algebraically defined upper and lower bounds $\lambda ^{+}(v)$ and $\lambda ^{-}(v)$ for the asymptotic behavior of the function $\frac{log|g(v)|}{d_{R}(g, G_{v})}$ as $d_{R}(g, G_{v}) \rightarrow \infty $ . The upper bound $\lambda ^{+}(v)$ is at most 1. The orbit $G(v)$ is closed in $\mathbb R ^{n} \Leftrightarrow \lambda ^{-}(w)$ is positive for some w $\in G(v)$ . If $G_{v}$ is compact, then $g \rightarrow |d_{R}(g,I) - d_{L}(g,I)|$ is uniformly bounded in $G$ , and the exponents $\lambda ^{+}(v)$ and $\lambda ^{-}(v)$ are sharp upper and lower asymptotic bounds for the functions $\frac{log|g(v)|}{d_{R}(g,I)}$ and $\frac{log|g(v)|}{d_{L}(g,I)}$ as $d_{R}(g,I) \rightarrow \infty $ or as $d_{L}(g,I) \rightarrow \infty $ . However, we show by example that if $G_{v}$ is noncompact, then there need not exist asymptotic upper and lower bounds for the function $\frac{log|g(v)|}{d_{L}(g, G_{v})}$ as $d_{L}(g, G_{v}) \rightarrow \infty $ . The results apply to representations of noncompact semisimple Lie groups $G$ on finite dimensional real vector spaces. We compute $\lambda ^{+}$ and $\lambda ^{-}$ for the irreducible, real representations of $SL(2,\mathbb R )$ , and we show that if the dimension of the $SL(2,\mathbb R )$ -module $V$ is odd, then $\lambda ^{+} = \lambda ^{-}$ on a nonempty open subset of $V$ . We show that the function $\lambda ^{-}$ is $K$ -invariant, where $K = O(n,\mathbb R ) \cap G$ . We do not know if $\lambda ^{-}$ is $G$ -invariant.     

Prolongations of valuations to finite extensions

Let ${K=\mathbb{Q}(\theta)}$ be an algebraic number field with θ in the ring A _K of algebraic integers of K and f(x) be the minimal polynomial of θ over the field ${\mathbb{Q}}$ of rational numbers. For a rational prime p, let ${\bar{f}(x)\,=\,\bar{g}_{1}(x)^{e_{1}}....\bar{g}_{r}(x)^{e_{r}}}$ be the factorization of the polynomial ${\bar{f}(x)}$ obtained by reducing coefficients of f(x) modulo p into a product of powers of distinct irreducible polynomials over ${\mathbb{Z}/p\mathbb{Z}}$ with g _i (x) monic. Dedekind proved that if p does not divide [ ${A_{K}:\mathbb{Z}}$ [θ]], then ${pA_{K}=\wp_{1}^{e_{1}}\ldots\wp_{r}^{e_{r}}}$ , where ${\wp_{1},\ldots,\wp_{r}}$ are distinct prime ideals of A _K , ${\wp_{i}=pA_{K}+g_{i}(\theta)A_{K}}$ having residual degree equal to the degree of ${\bar{g}_{i}(x)}$ . He also proved that p does not divide [ ${A_{K}:\mathbb{Z}}$ [θ]] if and only if for each i, either e _i  = 1 or ${\bar{g}_{i}(x)}$ does not divide ${\bar{M}(x)}$ where ${M(x)=\frac{1}{p}(f(x)-g_{1}(x)^{e_{1}}....g_{r}(x)^{e_{r}})}$ . Our aim is to give a weaker condition than the one given by Dedekind which ensures that if the polynomial ${\bar{f}(x)}$ factors as above over ${\mathbb{Z}/p\mathbb{Z}}$ , then there are exactly r prime ideals of A _K lying over p, with respective residual degrees ${\deg \bar {g}_{1}(x),...,\deg \bar {g}_{r}(x)}$ and ramification indices e ₁, ..., e _r . In this paper, the above problem has been dealt with in a more general situation when the base field is a valued field (K, v) of arbitrary rank and K(θ) is any finite extension of K.     

Constant Angle Property and Canonical Principal Directions for Surfaces in {\mathbb{M}^{2}(c) \times {\mathbb{R}_{1}}}

In this paper, we study surfaces in Lorentzian product spaces ${{\mathbb{M}^{2}(c) \times \mathbb{R}_1}}$ . We classify constant angle spacelike and timelike surfaces in ${{\mathbb{S}^{2} \times \mathbb{R}_1}}$ and ${{\mathbb{H}^{2} \times \mathbb{R}_1}}$ . Moreover, complete classifications of spacelike surfaces in ${{\mathbb{S}^{2} \times \mathbb{R}_1}}$ and ${{\mathbb{H}^{2} \times \mathbb{R}_1}}$ and timelike surfaces in ${{\mathbb{M}^{2}(c) \times \mathbb{R}_1}}$ with a canonical principal direction are obtained. Finally, a new characterization of the catenoid of the 3rd kind is established, as the only minimal timelike surface with a canonical principal direction in Minkowski 3–space.     

Saddle solutions for bistable symmetric semilinear elliptic equations

This paper concerns the existence and asymptotic characterization of saddle solutions in ${\mathbb {R}^{3}}$ for semilinear elliptic equations of the form $$-\Delta u + W'(u) = 0,\quad (x, y, z) \in {\mathbb {R}^{3}} \qquad\qquad\qquad (0.1)$$ where ${W \in \mathcal{C}^{3}(\mathbb {R})}$ is a double well symmetric potential, i.e. it satisfies W(?s) =  W(s) for ${s \in \mathbb {R},W(s) > 0}$ for ${s \in (-1,1)}$ , ${W(\pm 1) = 0}$ and ${W''(\pm 1) > 0}$ . Denoted with ${\theta_{2}}$ the saddle planar solution of (0.1), we show the existence of a unique solution ${\theta_{3} \in {\mathcal{C}^{2}}(\mathbb {R}^{3})}$ which is odd with respect to each variable, symmetric with respect to the diagonal planes, verifies ${0 < \theta_{3}(x,y,z) < 1}$ for x, y, z >  0 and ${\theta_{3}(x, y, z) \to_{z \to + \infty} \theta_{2}(x, y)}$ uniformly with respect to ${(x, y) \in \mathbb {R}^{2}}$ .     

The Left Hilbert BVP for h-Regular Functions in Clifford Analysis

Consider the real Clifford algebra ${\mathbb{R}_{0,n}}$ generated by e ₁, e ₂, . . . , e _n satisfying ${e_{i}e_{j} + e_{j}e_{i} = -2\delta_{ij} , i, j = 1, 2, . . . , n, e_{0}}$ is the unit element. Let ${\Omega}$ be an open set in ${\mathbb{R}^{n+1}}$ . u(x) is called an h-regular function in ${\Omega}$ if $$D_{x}u(x) + \widehat{u}(x)h = 0, \quad\quad (0.1)$$ where ${D_x = \sum\limits_{i=0}^{n} e_{i}\partial_{xi}}$ is the Dirac operator in ${\mathbb{R}^{n+1}}$ , and ${\widehat{u}(x) = \sum \limits_{A} (-1)^{\#A}u_{A}(x)e_{A}, \#A}$ denotes the cardinality of A and ${h = \sum\limits_{k=0}^{n} h_{k}e_{k}}$ is a constant paravector. In this paper, we mainly consider the Hilbert boundary value problem (BVP) for h-regular functions in ${\mathbb{R}_{+}^{n+1}}$ .     

Harmonic mappings and conformal minimal immersions of Riemann surfaces into {\mathbb {R}^{\rm N}}

We prove that for any open Riemann surface ${\mathcal{N}}$ , natural number N ≥ 3, non-constant harmonic map ${h:\mathcal{N} \to \mathbb{R}}$ ^N?2 and holomorphic 2-form ${\mathfrak{H}}$ on ${\mathcal{N}}$ , there exists a weakly complete harmonic map ${X=(X_j)_{j=1,\ldots,{\sc N}}:\mathcal{N} \to \mathbb{R}^{\sc N}}$ with Hopf differential ${\mathfrak{H}}$ and ${(X_j)_{j=3,\ldots,{\sc N}}=h.}$ In particular, there exists a complete conformal minimal immersion ${Y=(Y_j)_{j=1,\ldots,{\sc N}}:\mathcal{N} \to \mathbb{R}^{\sc N}}$ such that ${(Y_j)_{j=3,\ldots,{\sc N}}=h}$ . As some consequences of these results (1) there exist complete full non-decomposable minimal surfaces with arbitrary conformal structure and whose generalized Gauss map is non-degenerate and fails to intersect N hyperplanes of ${\mathbb{CP}^{{\sc N}-1}}$ in general position. (2) There exist complete non-proper embedded minimal surfaces in ${\mathbb{R}^{\sc N},}$ ${\forall\,{\sc N} >3 .}$      

Nash equilibria on topological semilattices

A function ${u : X \to \mathbb{R}}$ defined on a partially ordered set is quasi-Leontief if, for all ${x \in X}$ , the upper level set ${\{x\prime \in X : u(x\prime) \geq u(x)\}}$ has a smallest element; such an element is an efficient point of u. An abstract game ${u_{i} : \prod^{n}_{j=1} X_j \to \mathbb{R}, i \in \{1, \ldots , n\}}$ , is a quasi-Leontief game if, for all i and all ${(x_{j})_{j \neq i} \in \prod_{j \neq i} X_{j}, u_{i}((x_{j})_{j \neq i};-) : X_{i} \to \mathbb{R}}$ is quasi-Leontief; a Nash equilibrium x* of an abstract game ${u_{i} :\prod^{n}_{j=1} X_{j} \to \mathbb{R}}$ is efficient if, for all ${i, x^{*}_{i}}$ is an efficient point of the partial function ${u_{i}((x^{*}_{j})_{j \neq i};-) : X_{i} \to \mathbb{R}}$ . We establish the existence of efficient Nash equilibria when the strategy spaces X _i are topological semilattices which are Peano continua and Lawson semilattices.     

A class of subelliptic quasilinear equations

Suppose $\mathfrak {X} = \{X_1, X_2, \ldots,\,X_m\}$ is a system of real smooth vector fields on an open neighbourhood Ω of the closure of a bounded connected open set M in $\mathbb {R}^N$ satisfying the finite rank condition of Hörmander, namely the rank of the Lie algebra generated by $\mathfrak {X}$ under the usual bracket operation is a constant equal to N. We study the smoothness of solutions of a class of quasilinear equations of the form $$Q_{\mathfrak {X}}u = \sum _{j=1}^m X_j^*a_j(x, u, Xu) +b (x, u, Xu) = 0$$ where $a_j,\,b \in C^{\infty}(\Omega \times \mathbb {R} \times \mathbb {R}^m; \mathbb {R})$ . It is shown that if the matrix $\left({\frac {\partial a_j}{\partial \xi_i}}\right)$ is positive definite on $M \times \mathbb {R}^{m+1}$ then any weak solution $u \in \mathcal {C}^{2,\alpha}(M, \mathfrak {X})$ belongs to C ^∞(M).     

A decomposition of the universal embedding space for the near polygon {{\mathbb H}_n}

Let ${{\mathbb H}_n, n \geq 1}$ , be the near 2n-gon defined on the 1-factors of the complete graph on 2n?+?2 vertices, and let e denote the absolutely universal embedding of ${{\mathbb H}_n}$ into PG(W), where W is a ${\frac{1}{n+2} \left(\begin{array}{c}2n+2 \\ n+1\end{array}\right)}$ -dimensional vector space over the field ${{\mathbb F}_2}$ with two elements. For every point z of ${{\mathbb H}_n}$ and every ${i \in {\mathbb N}}$ , let Δ _i (z) denote the set of points of ${{\mathbb H}_n}$ at distance i from z. We show that for every pair {x, y} of mutually opposite points of ${{\mathbb H}_n, W}$ can be written as a direct sum ${W_0 \oplus W_1 \oplus \cdots \oplus W_n}$ such that the following four properties hold for every ${i \in \{0,\ldots,n \}}$ : (1) ${\langle e(\Delta_i(x) \cap \Delta_{n-i}(y)) \rangle = {\rm PG}(W_i)}$ ; (2) ${\left\langle e \left( \bigcup_{j \leq i} \Delta_j(x) \right) \right\rangle = {\rm PG}(W_0 \oplus W_1 \oplus \cdots \oplus W_i)}$ ; (3) ${\left\langle e \left( \bigcup_{j \leq i} \Delta_j(y) \right) \right\rangle = {\rm PG}(W_{n-i}\oplus W_{n-i+1} \oplus \cdots \oplus W_n)}$ ; (4) ${\dim(W_i) = |\Delta_i(x) \cap \Delta_{n-i}(y)| = \left(\begin{array}{c}n \\ i\end{array}\right)^2 - \left(\begin{array}{c}n \\ i-1\end{array}\right) \cdot \left(\begin{array}{c}n \\ i+1\end{array}\right)}$ .     

SIP: critical value functions have finite modulus of non-convexity

We consider semi-infinite programming problems ${{\rm SIP}(z)}$ depending on a finite dimensional parameter ${z \in \mathbb{R}^p}$ . Provided that ${\bar{x}}$ is a strongly stable stationary point of ${{\rm SIP}(\bar{z})}$ , there exists a locally unique and continuous stationary point mapping ${z \mapsto x(z)}$ . This defines the local critical value function ${\varphi(z) := f(x(z); z)}$ , where ${x \mapsto f(x; z)}$ denotes the objective function of ${{\rm SIP}(z)}$ for a given parameter vector ${z\in \mathbb{R}^p}$ . We show that ${\varphi}$ is the sum of a convex function and a smooth function. In particular, this excludes the appearance of negative kinks in the graph of ${\varphi}$ .     

Nikodym Maximal Functions Associated with Variable Planes in {\mathbb{R}^{3}}

Given a vector field ${\mathfrak{a}}$ on ${\mathbb{R}^3}$ , we consider a mapping ${x\mapsto \Pi_{\mathfrak{a}}(x)}$ that assigns to each ${x\in\mathbb{R}^3}$ , a plane ${\Pi_{\mathfrak{a}}(x)}$ containing x, whose normal vector is ${\mathfrak{a}(x)}$ . Associated with this mapping, we define a maximal operator ${\mathcal{M}^{\mathfrak{a}}_N}$ on ${L^1_{loc}(\mathbb{R}^3)}$ for each ${N\gg 1}$ by $$\mathcal{M}^{\mathfrak{a}}_Nf(x)=\sup_{x\in\tau} \frac{1}{|\tau|} \int_{\tau}|f(y)|\,dy$$ where the supremum is taken over all 1/N ×? 1/N?× 1 tubes τ whose axis is embedded in the plane ${\Pi_\mathfrak{a}(x)}$ . We study the behavior of ${\mathcal{M}^{\mathfrak{a}}_N}$ according to various vector fields ${\mathfrak{a}}$ . In particular, we classify the operator norms of ${\mathcal{M}^{\mathfrak{a}}_N}$ on ${L^2(\mathbb{R}^3)}$ when ${\mathfrak{a}(x)}$ is the linear function of the form (a ₁₁ x ₁?+?a ₂₁ x ₂, a ₁₂ x ₁?+?a ₂₂ x ₂, 1). The operator norm of ${\mathcal{M}^\mathfrak{a}_N}$ on ${L^2(\mathbb{R}^3)}$ is related with the number given by $$D=(a_{12}+a_{21})^2-4a_{11}a_{22}.$$      

Lax embeddings of the Hermitian unital

In this paper, we prove that every lax generalized Veronesean embedding of the Hermitian unital ${\mathcal{U}}$ of ${\mathsf{PG}(2,\mathbb{L}), \mathbb{L}}$ a quadratic extension of the field ${\mathbb{K}}$ and ${|\mathbb{K}| \geq 3}$ , in a ${\mathsf{PG}(d,\mathbb{F})}$ , with ${\mathbb{F}}$ any field and d ≥ 7, such that disjoint blocks span disjoint subspaces, is the standard Veronesean embedding in a subgeometry ${\mathsf{PG}(7,\mathbb{K}^{\prime})}$ of ${\mathsf{PG}(7,\mathbb{F})}$ (and d = 7) or it consists of the projection from a point ${p \in \mathcal{U}}$ of ${\mathcal{U}{\setminus} \{p\}}$ from a subgeometry ${\mathsf{PG}(7,\mathbb{K}^{\prime})}$ of ${\mathsf{PG}(7,\mathbb{F})}$ into a hyperplane ${\mathsf{PG}(6,\mathbb{K}^{\prime})}$ . In order to do so, when ${|\mathbb{K}| >3 }$ we strongly use the linear representation of the affine part of ${\mathcal{U}}$ (the line at infinity being secant) as the affine part of the generalized quadrangle ${\mathsf{Q}(4,\mathbb{K})}$ (the solid at infinity being non-singular); when ${|\mathbb{K}| =3}$ , we use the connection of ${\mathcal{U}}$ with the generalized hexagon of order 2.     

On the singular set of minimizers of p(x)-energies

We treat the partial regularity of locally bounded local minimizers $u$ for the $p(x)$ -energy functional $$\begin{aligned} \mathcal{E }(v;\Omega ) = \int \left( g^{\alpha \beta }(x)h_{ij}(v) D_\alpha v^i (x) D_\beta v^j (x) \right) ^{p(x)/2} dx, \end{aligned}$$ defined for maps $v : \Omega (\subset \mathbb R ^m) \rightarrow \mathbb R ^n$ . Assuming the Lipschitz continuity of the exponent $p(x) \ge 2$ , we prove that $u \in C^{1,\alpha }(\Omega _0)$ for some $\alpha \in (0,1)$ and an open set $\Omega _0 \subset \Omega $ with $\dim _\mathcal{H }(\Omega \setminus \Omega _0) \le m-[\gamma _1]-1$ , where $\dim _\mathcal{H }$ stands for the Hausdorff dimension, $[\gamma _1]$ the integral part of $\gamma _1$ , and $\gamma _1 = \inf p(x)$ .     

Existence of entire positive solutions for semilinear elliptic systems with gradient term

Under the Keller?COsserman condition on ${\Sigma_{j=1}^{2}f_{j}}$ , we show the existence of entire positive solutions for the semilinear elliptic system ${\Delta u_{1}+|\nabla u_{1}|=p_{1}(x)f_{1}(u_{1},u_{2}), \Delta u_{2}+|\nabla u_{2}|=p_{2}(x)f_{2}(u_{1},u_{2}),x \in \mathbb{R}^{N}}$ , where ${p_{j}(j=1, 2):\mathbb{R}^{N} \rightarrow [0,\infty)}$ are continuous functions.     

Pfaffian representations of cubic surfaces

Let $\mathbb{K }$ be a field of characteristic zero. We describe an algorithm which requires a homogeneous polynomial $F$ of degree three in $\mathbb{K }[x_{0},x_1,x_{2},x_{3}]$ and a zero ${\mathbf{a }}$ of $F$ in $\mathbb{P }^{3}_{\mathbb{K }}$ and ensures a linear Pfaffian representation of $\text{ V}(F)$ with entries in $\mathbb{K }[x_{0},x_{1},x_{2},x_{3}]$ , under mild assumptions on $F$ and ${\mathbf{a }}$ . We use this result to give an explicit construction of (and to prove the existence of) a linear Pfaffian representation of $\text{ V}(F)$ , with entries in $\mathbb{K }^{\prime }[x_{0},x_{1},x_{2},x_{3}]$ , being $\mathbb{K }^{\prime }$ an algebraic extension of $\mathbb{K }$ of degree at most six. An explicit example of such a construction is given.