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## Some bounds arising from a polynomial ideal associated to any $t$-design

#### William J. MARTIN [1] , Douglas R. STINSON [2]

We consider ordered pairs $(X,\mathcal{B})$ where $X$ is a finite set of size $v$ and $\mathcal{B}$ is some collection of $k$-element subsets of $X$ such that every $t$-element subset of $X$ is contained in exactly $\lambda$ blocks'' $B\in \mathcal{B}$ for some fixed $\lambda$. We represent each block $B$ by a zero-one vector $\bc_B$ of length $v$ and explore the ideal $\mathcal{I}(\mathcal{B})$ of polynomials in $v$ variables with complex coefficients which vanish on the set $\{ \bc_B \mid B \in \mathcal{B}\}$. After setting up the basic theory, we investigate two parameters related to this ideal: $\gamma_1(\mathcal{B})$ is the smallest degree of a non-trivial polynomial in the ideal $\mathcal{I}(\mathcal{B})$ and $\gamma_2(\mathcal{B})$ is the smallest integer $s$ such that $\mathcal{I}(\mathcal{B})$ is generated by a set of polynomials of degree at most $s$. We first prove the general bounds $t/2 < \gamma_1(\mathcal{B}) \le \gamma_2(\mathcal{B}) \le k$. Examining important families of examples, we find that, for symmetric 2-designs and Steiner systems, we have $\gamma_2(\mathcal{B}) \le t$. But we expect $\gamma_2(\mathcal{B})$ to be closer to $k$ for less structured designs and we indicate this by constructing infinitely many triple systems satisfying $\gamma_2(\mathcal{B})=k$.
Design, Steiner system, Polynomial ideal, Bounds
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Primary Language en Engineering Articles Orcid: 0000-0002-2027-5859Author: William J. MARTIN Institution: Worcester Polytechnic InstituteCountry: United States Orcid: 0000-0001-5635-8122Author: Douglas R. STINSON Institution: University of WaterlooCountry: Canada Publication Date : May 7, 2020
 Bibtex @research article { jacodesmath729446, journal = {Journal of Algebra Combinatorics Discrete Structures and Applications}, issn = {}, eissn = {2148-838X}, address = {}, publisher = {Yildiz Technical University}, year = {2020}, volume = {7}, pages = {161 - 181}, doi = {10.13069/jacodesmath.729446}, title = {Some bounds arising from a polynomial ideal associated to any \$t\$-design}, key = {cite}, author = {J. Martın, William and R. Stınson, Douglas} } APA J. Martın, W , R. Stınson, D . (2020). Some bounds arising from a polynomial ideal associated to any $t$-design . Journal of Algebra Combinatorics Discrete Structures and Applications , 7 (2) , 161-181 . DOI: 10.13069/jacodesmath.729446 MLA J. Martın, W , R. Stınson, D . "Some bounds arising from a polynomial ideal associated to any $t$-design" . Journal of Algebra Combinatorics Discrete Structures and Applications 7 (2020 ): 161-181 Chicago J. Martın, W , R. Stınson, D . "Some bounds arising from a polynomial ideal associated to any $t$-design". Journal of Algebra Combinatorics Discrete Structures and Applications 7 (2020 ): 161-181 RIS TY - JOUR T1 - Some bounds arising from a polynomial ideal associated to any $t$-design AU - William J. Martın , Douglas R. Stınson Y1 - 2020 PY - 2020 N1 - doi: 10.13069/jacodesmath.729446 DO - 10.13069/jacodesmath.729446 T2 - Journal of Algebra Combinatorics Discrete Structures and Applications JF - Journal JO - JOR SP - 161 EP - 181 VL - 7 IS - 2 SN - -2148-838X M3 - doi: 10.13069/jacodesmath.729446 UR - https://doi.org/10.13069/jacodesmath.729446 Y2 - 2019 ER - EndNote %0 Journal of Algebra Combinatorics Discrete Structures and Applications Some bounds arising from a polynomial ideal associated to any $t$-design %A William J. Martın , Douglas R. Stınson %T Some bounds arising from a polynomial ideal associated to any $t$-design %D 2020 %J Journal of Algebra Combinatorics Discrete Structures and Applications %P -2148-838X %V 7 %N 2 %R doi: 10.13069/jacodesmath.729446 %U 10.13069/jacodesmath.729446 ISNAD J. Martın, William , R. Stınson, Douglas . "Some bounds arising from a polynomial ideal associated to any $t$-design". Journal of Algebra Combinatorics Discrete Structures and Applications 7 / 2 (May 2020): 161-181 . https://doi.org/10.13069/jacodesmath.729446 AMA J. Martın W , R. Stınson D . Some bounds arising from a polynomial ideal associated to any $t$-design. Journal of Algebra Combinatorics Discrete Structures and Applications. 2020; 7(2): 161-181. Vancouver J. Martın W , R. Stınson D . Some bounds arising from a polynomial ideal associated to any $t$-design. Journal of Algebra Combinatorics Discrete Structures and Applications. 2020; 7(2): 161-181.

Authors of the Article
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