Polynomials as terms and the Boolean Independence Theorem
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866911766090350592 |
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| author | Klazar, M. |
| author_facet | Klazar, M. |
| contents | We develop a theory of formal multivariate polynomials over commutative rings by treating them as ring terms. Our main result is that two ring terms are s-equivalent (when expanded they yield the same standard polynomial) iff they are f-equivalent (one can be transformed in the other by a series of elementary transformations). We consider in a similar way Boolean terms (formulas) and prove a theorem that two events $a$ and $b$ in a probability space, which are built by two Boolean terms from respective tuples $A$ and $B$ of elementary events, are independent if the events in $A$ are independent of the events in $B$. This theorem rigorizes arguments in the Probabilistic Method in Combinatorics. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_10033 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Polynomials as terms and the Boolean Independence Theorem Klazar, M. Combinatorics Commutative Algebra Logic We develop a theory of formal multivariate polynomials over commutative rings by treating them as ring terms. Our main result is that two ring terms are s-equivalent (when expanded they yield the same standard polynomial) iff they are f-equivalent (one can be transformed in the other by a series of elementary transformations). We consider in a similar way Boolean terms (formulas) and prove a theorem that two events $a$ and $b$ in a probability space, which are built by two Boolean terms from respective tuples $A$ and $B$ of elementary events, are independent if the events in $A$ are independent of the events in $B$. This theorem rigorizes arguments in the Probabilistic Method in Combinatorics. |
| title | Polynomials as terms and the Boolean Independence Theorem |
| topic | Combinatorics Commutative Algebra Logic |
| url | https://arxiv.org/abs/2401.10033 |