arXiv subjects

Tetsuya Ando

Publications and source records attributed to Tetsuya Ando.

3 recordsLinked to original sources

arXiv2023

Extremal Cubic Inequalities of Three Variables

Let $\mathcal{H}_{3,d}$ be the vector space of homogeneous three variable polynomials of degree $d$, and $\mathcal{P}_{3,d}^+$ be the set of all elements $f \in \mathcal{H}_{3,d}$ such that$f(x,y,z) \geq 0$ for all $x \geq 0$, $y \geq 0$, $z \geq 0$. In this article, we determine all extremal elements of $\mathcal{P}_{3,3}^+$. We prove that if $f \in \mathcal{P}_{3,3}^+$ is an irreducible extremal element, then the zero locus $V_{\mathbb{C}}(f)$ in $\mathbb{P}_{\mathbb{C}}^2$ is a rational curve whose singularity is an acnode in the interior of $\mathbb{P}_+^2$ or a cusp on an edge of $\mathbb{P}_+^2$. We also prove that if $f \in \mathcal{P}_{3,3}^+$ is an extremal element, then $f(x^2,y^2,z^2)$ is an extremal element of $\mathcal{P}_{3,6}$, where $\mathcal{P}_{3,d}$ is the set of all the elements $f \in \mathcal{H}_{3,d}$ such that $f(x,y,z) \geq 0$ for all $x$, $y$, $z \in \mathbb{R}$. A notion of infinitely near zeros of an inequality is introduced, and plays an important role.

math.AG↗

arXiv2022

Some Extremal Symmetric Inequalities

Let $\mathcal{H}_{n,d} := \mathbb{R}[x_1$,$\ldots$, $x_n]_d$ be the set of all the homogeneous polynomials of degree $d$, and let $\mathcal{H}_{n,d}^s := \mathcal{H}_{n,d}^{\mathfrak{S}_n}$ be the subset of all the symmetric polynomials. For a semialgebraic subset of $A \subset \mathbb{R}^n$ and a vector subspace $\mathcal{H} \subset \mathcal{H}_{n,d}$, we define a PSD cone $\mathcal{P}(A$, $\mathcal{H})$ by $\mathcal{P}(A$, $\mathcal{H}) := \big\{f \in \mathcal{H}$ $\big|$ $f(a) \geq 0$ ($\forall a \in A$)$\big\}$. In this article, we study a family of extremal symmetric polynomials of $\mathcal{P}_{3,6} := \mathcal{P}(\mathbb{R}^3$, $\mathcal{H}_{3,6})$ and that of $\mathcal{P}_{4,4} := \mathcal{P}(\mathbb{R}^4$, $\mathcal{H}_{4,4})$. We also determine all the extremal polynomials of $\mathcal{P}_{3,5}^{s+} := \mathcal{P}(\mathbb{R}_+^3$, $\mathcal{H}_{3,5}^s)$ where $\mathbb{R}_+ := \big\{ x \in \mathbb{R}$, $x \geq 0 \big\}$. Some of them provide extremal polynomials of $\mathcal{P}_{3,10}$.

math.AG↗

arXiv2022

Some Cubic and Quartic Inequalities of Four Variables

Let $\mathcal{H} \subset \mathcal{H}_{n,d} := \mathbb{R}[x_1$,$\ldots$, $x_n]_d$ be a vector space, and $A$ be a compact semialgebraic subset of $\mathbb{P}_{\mathbb{R}}^{n-1}$. We shall study some PSD cones $\mathcal{P} = \mathcal{P}(A$, $\mathcal{H}) := \big\{f \in \mathcal{H}$ $\big|$ $f(a) \geq 0$ ($\forall a \in A$)$\big\}$. Our interests are (1) to determine the extremal elements of $\mathcal{P}$, (2) to determine discriminants of $\mathcal{P}$, (3) to describe $\mathcal{P}$ as a union of basic semialgebraic subsets, and (4) to find a nice test set when $\dim \mathcal{H}$ is low. In this article, we present (1), (2), (3) and (4) for $\mathcal{P}(\mathbb{R}^4$, $\mathcal{H}_{4,4}^{s0})$ and $\mathcal{P}(\mathbb{R}_+^4$, $\mathcal{H}_{4,4}^{s0})$, where $\mathcal{H}_{n,d}^{s0} := \big\{f \in \mathcal{H}_{n,d}$ $\big|$ $f$ is symmetric and $f(1,\ldots,1)=0 \big\}$. We also provide (1) -- (4) for $\mathcal{P}(\mathbb{R}_+^4$, $\mathcal{H}_{4,3}^{c0})$, where $\mathcal{H}_{n,d}^{c0} := \big\{f \in \mathcal{H}_{n,d}$ $\big|$ $f$ is cyclic and $f(1,\ldots,1)=0 \big\}$.

math.AG↗