arXiv subjects

Kohhei Yamaguchi

Publications and source records attributed to Kohhei Yamaguchi.

14 recordsLinked to original sources

arXiv2026

Homotopy stability for spaces of triples of real polynomials without common roots

We continue our study of the topology of the spaces of $m$ tuples of real polynomials with common degree $d$ and without common roots of multiplicity $n$, and in particular their stability properties with respect to $d$. In an earlier paper we have proved a homotopy stability result and determined the stable homotopy types of such spaces in the case $m n >=4$. In the case $m n= 3$ we could only prove stability in homology. In this paper we settle the case (m,n) = (3,1): the space of triples of monic real polynomials of the same degree having no common root. We prove homotopy stability and determine the precise stability range supplied by our method. We also show that in degree 2, the distinction between homotopy equivalence up to and through the stability dimension is essential. The other borderline case (m,n) = (1,3) is not treated here. We expect that homotopy stability will hold also in that case and hope to return to it in future work.

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arXiv2024

Spaces of non-resultant systems of real bounded multiplicity determined by a toric variety

For any field $\Bbb F$ and positive integers $m,n,d$ with $(m,n)\not= (1,1)$, Farb and Wolfson defined the certain affine varieties ${\rm Poly}^{d,m}_n(\Bbb F)$ as generalizations of spaces first studied by Arnold, Vassiliev, Segal and others. As a natural generalization of this, for each fan $Σ$ and $r$-tuple $D=(d_1,\cdots ,d_r)$ of positive integers, the current authors defined spaces ${\rm Poly}^{D,Σ}_n(\Bbb F)$, where $r$ is the number of one dimensional cones in $Σ$. These spaces can also be regarded as generalizations of the space ${\rm Hol}^*_D(S^2,X_Σ)$ of based rational curves from the Riemann sphere $S^2$ to the toric variety $X_Σ$ of degree $D$, where $X_Σ$ denotes the toric variety (over $\Bbb C$) corresponding to the fan $Σ$. In this paper, we define spaces ${\rm Q}^{D,Σ}_n(\Bbb F)$ ($\Bbb F=\Bbb R$ or $\Bbb C$) which are real analogues of ${\rm Poly}^{D,Σ}_n(\Bbb F)$ and which can be viewed as a generalizations of spaces considered by Arnold, Vassiliev and others in the context of real singularity theory. We prove that homotopy stability holds for these spaces and compute the stability dimensions explicitly.

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arXiv2022

Spaces of non-resultant systems of bounded multiplicity with real coefficients

For each pair $(m,n)$ of positive integers with $(m,n)\not= (1,1)$ and an arbitrary field $\bf F$ with algebraic closure $\overline{\bf F}$, let $\rm Po^{d,m}_n(\bf F)$ denote the space of $m$-tuples $(f_1(z),\cdots ,f_m(z))\in \bf F [z]^m$ of $\bf F$-coefficients monic polynomials of the same degree $d$ such that the polynomials $\{f_k(z)\}_{k=1}^m$ have no common root in $\overline{\bf F}$ of multiplicity $\geq n$. These spaces $\rm Po^{d,m}_n(\bf F)$ were first defined and studied by B. Farb and J. Wolfson as generalizations of spaces first studied by Arnold, Vassiliev and Segal and others in several different contexts. In previous we determined explicitly the homotopy type of this space in the case $\bf F =\Bbb C$. In this paper, we investigate the case $\bf F =\Bbb R$.

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arXiv2021

Spaces of non-resultant systems of bounded multiplicity determined by a toric variety

The space of non-resultant systems of bounded multiplicity for a toric variety X is a generalization of the space of rational curves on it. In our earlier work we proved a homotopy stability theorem and determined explicitly the homotopy type of this space for the case X = CP^m. In this paper we consider the case of a general non-singular toric variety and prove a homotopy stability theorem generalising the one for CP^m.

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arXiv2021

The homotopy type of spaces of real resultants with bounded multiplicity

For positive integers $d,m,n\geq 1$ with $(m,n)\not= (1,1)$ and $\Bbb K=\Bbb R$ or $\Bbb C$, let $Q^{d,m}_{n}(\Bbb K)$ denote the space of $m$-tuples $(f_1(z),\cdots ,f_m(z))\in \Bbb K [z]^m$ of $\Bbb K$-coefficients monic polynomials of the same degree $d$ such that polynomials $\{f_k(z)\}_{k=1}^m$ have no common {\it real} root of multiplicity $\geq n$ (but may have complex common root of any multiplicity). %% These spaces can be regarded as one of generalizations of the spaces defined and studied by Arnold and Vassiliev, and they may be also considered as the real analogues of the spaces studied by B. Farb and J. Wolfson. In this paper, we shall determine their homotopy types explicitly and generalize some previously obtained results.

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arXiv2020

The homotopy type of the space of algebraic loops on a toric variety

We investigate the homotopy type of the space of tuples of polynomials inducing base-point preserving algebraic maps from the circle S1 to a toric variety XΣ. In particular, we prove a homotopy stability result for this space by combining the Vassiliev spectral sequence and the scanning map.

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arXiv2017

The homotopy type of spaces of rational curves on a toric variety

Spaces of holomorphic maps from the Riemann sphere to various complex manifolds (holomorphic curves ) have played an important role in several area of mathematics. In a seminal paper G. Segal investigated the homotopy type of holomorphic curves on complex projective spaces and M. Guest on compact smooth toric varieties.. Recently Mostovoy and Villanueva, obtained a far reaching generalisation of these results, and in particular (for holomorphic curves) improved the stability dimension obtained by Guest. In this paper, we generalize their result to holomorphic curves, on certain non-compact smooth toric varieties.

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arXiv2016

The homotopy type of spaces of resultants of bounded multiplicity

For positive integers $m,n, d\geq 1$ with $(m,n)\not= (1,1)$ and a field $\Bbb F$ with its algebraic closure $\overline{\Bbb F}$, let $\text{Poly}^{d,m}_n(\Bbb F)$ denote the space of all $m$-tuples $(f_1(z),\cdots ,f_m(z))\in \Bbb F [z]$ of monic polynomials of the same degree $d$ such that polynomials $f_1(z),\cdots ,f_m(z)$ have no common root in $\overline{\Bbb F}$ of multiplicity $\geq n$. These spaces were defined by Farb and Wolfson in \cite{FW} as generalizations of spaces first studied by Arnold, Vassiliev, Segal and others in different contexts. In \cite{FW} they obtained algebraic geometrical and arithmetic results about the topology of these spaces. In this paper we investigate the homotopy type of these spaces for the case $\Bbb F =\mathbb{C}$. Our results generalize those of \cite{FW} for $\Bbb F =\Bbb C$ and also results of G. Segal \cite{Se}, V. Vassiliev \cite{Va} and F.Cohen-R.Cohen-B.Mann-R.Milgram \cite{CCMM} for $m\geq 2$ and $n\geq 2$.

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arXiv2014

The homotopy type of spaces of coprime polynomials revisited

The purpose of this paper is to study the topology of certain toric varieties $X_I$, arising as quotients of the action of $\C^*$ on complements of arrangements of coordinate subspaces in $\C^n$, and to improve the homotopy stability dimension for the inclusion map $i_d:\Hol_d^*(S^2,X_I)\to \Map_d^*(S^2,X_I)$ given in \cite{GKY1} by using spectral sequences induced from simplicial resolutions.

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arXiv2014

Spaces of algebraic maps from real projective spaces to toric varieties

The problem of approximating the infinite dimensional space of all continuous maps from an algebraic variety $X$ to an algebraic variety $Y$ by finite dimensional spaces of algebraic maps arises in several areas of geometry and mathematical physics. An often considered formulation of the problem (sometimes called the Atiyah-Jones problem after \cite{AJ}) is to determine a (preferably optimal) integer $n_D$ such that the inclusion from this finite dimensional algebraic space into the corresponding infinite dimensional one induces isomorphisms of homology (or homotopy) groups through dimension $n_D$, where $D$ denotes a tuple of integers called the "degree" of the algebraic maps and $n_D\to\infty$ as $D\to\infty$. In this paper we investigate this problem in the case when $X$ is a real projective space and $Y$ is a smooth compact toric variety.

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arXiv2011

Simplicial resolutions and spaces of algebraic maps between real projective spaces

We show that the space $\tilde{A}_{d}(m,n)$ consisting of all real projective classes of $(n+1)$-tuples of real coefficients homogeneous polynomials of degree $d$ in $(m+1)$ variables, without common real roots except zero, has the same homology as the space $ \Map(\RP^m,\Bbb \RP^n)$ of continuous maps from the $m$-dimensional real projective space $\RP^m$ into the $n$ real dimensional projective space $\RP^n$ up to dimension %in dimensions smaller than $(n-m)(d+1)-1$. This considerably improves the main result of \cite{AKY1}.

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arXiv2011

Spaces of equivariant algebraic maps from real projective spaces into complex projective spaces

We study the homotopy types of certain spaces closely related to the spaces of algebraic (rational) maps from the $m$ dimensional real projective space into the $n$ dimensional complex projective space for $2\leq m\leq 2n$ (we conjecture this relation to be a homotopy equivalence). In an earlier article we proved that the homotopy types of the terms of the natural degree filtration approximate closer and closer the homotopy type of the space of continuous maps and obtained bounds that describe the closeness of the approximation in terms of the degrees of the maps. Here we improve the estimates of the bounds by using new methods introduced in \cite{Mo3} and used in \cite{KY4}. In addition, in the the last section, we prove a special case ($m=1$) of the conjecture stated in \cite{AKY1} that our spaces are homotopy equivalent to the spaces of algebraic maps.

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arXiv2010

Spaces of algebraic and continuous maps between real algebraic varieties

We consider the inclusion of the space of algebraic (regular) maps between real algebraic varieties in the space of all continuous maps. For a certain class of real algebraic varieties, which include real projective spaces, it is well known that the space of real algebraic maps is a dense subset of the space of all continuous maps. Our first result shows that, for this class of varieties, the inclusion is also a homotopy equivalence. After proving this, we restrict the class of varieties to real projective spaces. In this case, the space of algebraic maps has a ` minimum degree\rq filtration by finite dimensional subspaces and it is natural to expect that the homotopy types of the terms of the filtration approximate closer and closer the homotopy type of the space of continuous mappings as the degree increases. We prove this and compute the lower bounds of this approximation for ` even\rq components of these spaces (more precisely, we prove a very similar and closely related result, and state this one as a conjecture).

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arXiv2010

Spaces of algebraic maps from real projective spaces into complex projective spaces

We study the homotopy types of spaces of algebraic (rational) maps from real projective spaces into complex projective spaces. In a previous paper we have shown that the inclusion of the first space into the second one is a homotopy equivalence. In this paper we prove that the homotopy types of the terms of the natural "degree" filtration approximate closer and closer the homotopy type of the space of continuous maps and obtain bounds that describe the closeness of the approximation in terms of the degree. Moreover, we compute the homotopy groups of the spaces in low dimensions.

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