Generic forms
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866912632065228800 |
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| author | Fröberg, Ralf Löfwall, Clas |
| author_facet | Fröberg, Ralf Löfwall, Clas |
| contents | We study forms $I=(f_1,\ldots,f_r)$, $°f_i=d_i$, in $F$ which is the free associative algebra $k\langle x_1,\ldots,x_n\rangle$ or the polynomial ring $k[x_1,\ldots,x_n]$, where $k$ is a field and $°x_i=1$ for all $i$. We say that $I$ has type $t=(n;d_1,\ldots,d_r)$ and also that $F/I$ is a $t$-presentation. For each prime field $k_0$ and type $t=(n;d_1,\ldots,d_r)$, there is a series which is minimal among all Hilbert series for $t$-presentations over fields with prime field $k_0$ and such a $t$-presentation is called generic if its Hilbert series coincides with the minimal one. When the field is the real or complex numbers, we show that a $t$-presentation is generic if and only if
it belongs to a non-empty countable intersection $C$ of Zariski open subsets of the affine space, defined by the coefficients in the relations, such that all points in $C$ have the same Hilbert series.
In the commutative case there is a conjecture on what this minimal series is, and we give a conjecture for the generic series in the non-commutative quadratic case (building on work by Anick). We prove that if $A=k\langle x_1,\ldots,x_n\rangle/(f_1,\ldots,f_r)$ is a generic quadratic presentation, then $\{ x_if_j\}$ either is linearly independent or generate $A_3$. This complements a similar theorem by Hochster-Laksov in the commutative case.
Finally we show, a bit to our surprise, that the Koszul dual of a generic presentation is not generic in general. But if the relations have algebraically independent coefficients over the prime field, we prove that the Koszul dual is generic. Hereby, we give a counterexample of \cite[Proposition 4.2]{P-P}, which states a criterion for a generic non-commutative quadratic presentation to be Koszul. We formulate and prove a correct version of the proposition. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_13591 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Generic forms Fröberg, Ralf Löfwall, Clas Commutative Algebra We study forms $I=(f_1,\ldots,f_r)$, $°f_i=d_i$, in $F$ which is the free associative algebra $k\langle x_1,\ldots,x_n\rangle$ or the polynomial ring $k[x_1,\ldots,x_n]$, where $k$ is a field and $°x_i=1$ for all $i$. We say that $I$ has type $t=(n;d_1,\ldots,d_r)$ and also that $F/I$ is a $t$-presentation. For each prime field $k_0$ and type $t=(n;d_1,\ldots,d_r)$, there is a series which is minimal among all Hilbert series for $t$-presentations over fields with prime field $k_0$ and such a $t$-presentation is called generic if its Hilbert series coincides with the minimal one. When the field is the real or complex numbers, we show that a $t$-presentation is generic if and only if it belongs to a non-empty countable intersection $C$ of Zariski open subsets of the affine space, defined by the coefficients in the relations, such that all points in $C$ have the same Hilbert series. In the commutative case there is a conjecture on what this minimal series is, and we give a conjecture for the generic series in the non-commutative quadratic case (building on work by Anick). We prove that if $A=k\langle x_1,\ldots,x_n\rangle/(f_1,\ldots,f_r)$ is a generic quadratic presentation, then $\{ x_if_j\}$ either is linearly independent or generate $A_3$. This complements a similar theorem by Hochster-Laksov in the commutative case. Finally we show, a bit to our surprise, that the Koszul dual of a generic presentation is not generic in general. But if the relations have algebraically independent coefficients over the prime field, we prove that the Koszul dual is generic. Hereby, we give a counterexample of \cite[Proposition 4.2]{P-P}, which states a criterion for a generic non-commutative quadratic presentation to be Koszul. We formulate and prove a correct version of the proposition. |
| title | Generic forms |
| topic | Commutative Algebra |
| url | https://arxiv.org/abs/2504.13591 |