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| Format: | Preprint |
| Published: |
2023
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| Online Access: | https://arxiv.org/abs/2310.11619 |
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| _version_ | 1866929716238221312 |
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| author | Camphire, Heath |
| author_facet | Camphire, Heath |
| contents | Let $k$ be a field of odd characteristic $p$. Fix an even number $d<p+1$ and a power $q\geq d+3$ of $p$. For most choices of degree $d$ standard graded hypersurfaces $R=k[x,y,z]/(f)$ with homogeneous maximal ideal $\mathfrak{m}$, we can determine the graded Betti numbers of $R/\mathfrak{m}^{[q]}$. In fact, given two fixed powers $q_0,q_1\geq d+3$, for most choices of $R$ the graded Betti numbers in high homological degree of $R/\mathfrak{m}^{[q_0]}$ and $R/\mathfrak{m}^{[q_1]}$ are the same up to a constant shift. This thesis shows this fact by combining our results with the work of Miller, Rahmati, and R.G. on link-$q$-compressed polynomials in Betti numbers of the frobenius powers of the maximal ideal over certain hypersurfaces. We show that link-$q$-compressed polynomials are indeed fairly common in many polynomial rings. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_11619 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Determining the Betti numbers of $R/(x^{p^e},y^{p^e},z^{p^e})$ for most even degree hypersurfaces in odd characteristic Camphire, Heath Commutative Algebra Let $k$ be a field of odd characteristic $p$. Fix an even number $d<p+1$ and a power $q\geq d+3$ of $p$. For most choices of degree $d$ standard graded hypersurfaces $R=k[x,y,z]/(f)$ with homogeneous maximal ideal $\mathfrak{m}$, we can determine the graded Betti numbers of $R/\mathfrak{m}^{[q]}$. In fact, given two fixed powers $q_0,q_1\geq d+3$, for most choices of $R$ the graded Betti numbers in high homological degree of $R/\mathfrak{m}^{[q_0]}$ and $R/\mathfrak{m}^{[q_1]}$ are the same up to a constant shift. This thesis shows this fact by combining our results with the work of Miller, Rahmati, and R.G. on link-$q$-compressed polynomials in Betti numbers of the frobenius powers of the maximal ideal over certain hypersurfaces. We show that link-$q$-compressed polynomials are indeed fairly common in many polynomial rings. |
| title | Determining the Betti numbers of $R/(x^{p^e},y^{p^e},z^{p^e})$ for most even degree hypersurfaces in odd characteristic |
| topic | Commutative Algebra |
| url | https://arxiv.org/abs/2310.11619 |