Elimination by Substitution
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913260736872448 |
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| author | Kreuzer, Martin Robbiano, Lorenzo |
| author_facet | Kreuzer, Martin Robbiano, Lorenzo |
| contents | Let $K$ be a field and $P=K[x_1,\dots,x_n]$. The technique of elimination by substitution is based on discovering a coherently $Z=(z_1,\dots,z_s)$-separating tuple of polynomials $(f_1,\dots,f_s)$ in an ideal $I$, i.e., on finding polynomials such that $f_i = z_i - h_i$ with $h_i \in K[X \setminus Z]$. Here we elaborate on this technique in the case when $P$ is non-negatively graded. The existence of a coherently $Z$-separating tuple is reduced to solving several $P_0$-module membership problems. Best separable re-embeddings, i.e., isomorphisms $P/I \longrightarrow K[X \setminus Z] / (I \cap K[X \setminus Z])$ with maximal $\#Z$, are found degree-by-degree. They turn out to yield optimal re-embeddings in the positively graded case. Viewing $P_0 \longrightarrow P/I$ as a fibration over an affine space, we show that its fibers allow optimal $Z$-separating re-embeddings, and we provide a criterion for a fiber to be isomorphic to an affine space. In the last section we introduce a new technique based on the solution of a unimodular matrix problem which enables us to construct automorphisms of $P$ such that additional $Z$-separating re-embeddings are possible. One of the main outcomes is an algorithm which allows us to explicitly compute a homogeneous isomorphism between $P/I$ and a non-negatively graded polynomial ring if $P/I$ is regular. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_06415 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Elimination by Substitution Kreuzer, Martin Robbiano, Lorenzo Commutative Algebra 13P10 (Primary) 14Q20, 14R10, 14R25, 13C13 (Secondary) Let $K$ be a field and $P=K[x_1,\dots,x_n]$. The technique of elimination by substitution is based on discovering a coherently $Z=(z_1,\dots,z_s)$-separating tuple of polynomials $(f_1,\dots,f_s)$ in an ideal $I$, i.e., on finding polynomials such that $f_i = z_i - h_i$ with $h_i \in K[X \setminus Z]$. Here we elaborate on this technique in the case when $P$ is non-negatively graded. The existence of a coherently $Z$-separating tuple is reduced to solving several $P_0$-module membership problems. Best separable re-embeddings, i.e., isomorphisms $P/I \longrightarrow K[X \setminus Z] / (I \cap K[X \setminus Z])$ with maximal $\#Z$, are found degree-by-degree. They turn out to yield optimal re-embeddings in the positively graded case. Viewing $P_0 \longrightarrow P/I$ as a fibration over an affine space, we show that its fibers allow optimal $Z$-separating re-embeddings, and we provide a criterion for a fiber to be isomorphic to an affine space. In the last section we introduce a new technique based on the solution of a unimodular matrix problem which enables us to construct automorphisms of $P$ such that additional $Z$-separating re-embeddings are possible. One of the main outcomes is an algorithm which allows us to explicitly compute a homogeneous isomorphism between $P/I$ and a non-negatively graded polynomial ring if $P/I$ is regular. |
| title | Elimination by Substitution |
| topic | Commutative Algebra 13P10 (Primary) 14Q20, 14R10, 14R25, 13C13 (Secondary) |
| url | https://arxiv.org/abs/2403.06415 |