Factorizations of polynomials with integral non-negative coefficients
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866909581821607936 |
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| author | Campanini, Federico Facchini, Alberto |
| author_facet | Campanini, Federico Facchini, Alberto |
| contents | We study the structure of the commutative multiplicative monoid $\mathbb N_0[x]^*$ of all the non-zero polynomials in $\mathbb Z[x]$ with non-negative coefficients. We show that $\mathbb N_0[x]^*$ is not a half-factorial monoid and is not a Krull monoid, but has a structure very similar to that of Krull monoids, replacing valuations into $\mathbb N_0$ with derivations into $\mathbb N_0$. We study ideals, chain of ideals, prime ideals and prime elements of $\mathbb N_0[x]^*$. Our monoid $\mathbb N_0[x]^*$ is a submonoid of the multiplicative monoid of the ring $\mathbb Z[x]$, which is a left module over the Weyl algebra $A_1(\mathbb Z)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_12145 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Factorizations of polynomials with integral non-negative coefficients Campanini, Federico Facchini, Alberto Commutative Algebra We study the structure of the commutative multiplicative monoid $\mathbb N_0[x]^*$ of all the non-zero polynomials in $\mathbb Z[x]$ with non-negative coefficients. We show that $\mathbb N_0[x]^*$ is not a half-factorial monoid and is not a Krull monoid, but has a structure very similar to that of Krull monoids, replacing valuations into $\mathbb N_0$ with derivations into $\mathbb N_0$. We study ideals, chain of ideals, prime ideals and prime elements of $\mathbb N_0[x]^*$. Our monoid $\mathbb N_0[x]^*$ is a submonoid of the multiplicative monoid of the ring $\mathbb Z[x]$, which is a left module over the Weyl algebra $A_1(\mathbb Z)$. |
| title | Factorizations of polynomials with integral non-negative coefficients |
| topic | Commutative Algebra |
| url | https://arxiv.org/abs/2504.12145 |