arXiv subjects

Robert M. Walker

Publications and source records attributed to Robert M. Walker.

6 recordsLinked to original sources

arXiv2018

Tensor-Multinomial Sums of Ideals: Primary Decompositions and Persistence of Associated Primes

Given a polynomial ring $C$ over a field and proper ideals $I$ and $J$ whose generating sets involve disjoint variables, we determine how to embed the associated primes of each power of $I+J$ into a collection of primes described in terms of the associated primes of select powers of $I$ and of $J$. We record two applications. First, in case the field is algebraically closed, we construct primary decompositions for powers of $I+J$ from primary decompositions for powers of $I$ and $J$. Separately, we attack the persistence problem for associated primes of powers of an ideal in case one of $I$ or $J$ is a non-zero normal ideal.

math.AC↗

arXiv2018

Uniform Harbourne-Huneke Bounds via Flat Extensions

Over an arbitrary field $\mathbb{F}$, Harbourne conjectured that $$I^{(N (r-1)+1)} \subseteq I^r$$ for all $r>0$ and all homogeneous ideals $I$ in $S = \mathbb{F} [\mathbb{P}^N] = \mathbb{F} [x_0, \ldots, x_N]$. The conjecture has been disproven for select values of $N \ge 2$: first by Dumnicki, Szemberg, and Tutaj-Gasińska in characteristic zero, and then by Harbourne and Seceleanu in odd positive characteristic. However, the ideal containments above do hold when, for instance, $I$ is a monomial ideal in $S$. As a sequel to (arXiv:1510.02993), we present criteria for containments of type $I^{(N (r-1)+1)} \subseteq I^r$ for all $r>0$ and certain classes of ideals $I$ in a prodigious class of normal rings. Of particular interest is a result for monomial primes in tensor products of affine semigroup rings. Indeed, we explain how to give effective multipliers $N$ in several cases including: the $D$-th Veronese subring of any polynomial ring $\mathbb{F} [x_1, \ldots, x_n]$ $(n \ge 1)$; and the extension ring $\mathbb{F} [x_1, \ldots, x_n, z]/(z^D - x_1 \cdots x_n)$ of $\mathbb{F}[x_1, \ldots, x_n]$.

math.AC↗

arXiv2017

Uniform Symbolic Topologies via Multinomial Expansions

When does a Noetherian commutative ring $R$ have uniform symbolic topologies on primes--read, when does there exist an integer $D>0$ such that the symbolic power $P^{(Dr)} \subseteq P^r$ for all prime ideals $P \subseteq R$ and all $r >0$? Groundbreaking work of Ein-Lazarsfeld-Smith, as extended by Hochster and Huneke, and by Ma and Schwede in turn, provides a beautiful answer in the setting of finite-dimensional excellent regular rings. It is natural to then sleuth for analogues where the ring $R$ is non-regular, or where the above ideal containments can be improved using a linear function whose growth rate is slower. This manuscript falls under the overlap of these research directions. Working with a prescribed type of prime ideal $Q$ inside of tensor products of domains of finite type over an algebraically closed field $\mathbb{F}$, we present binomial- and multinomial expansion criteria for containments of type $Q^{(E r)} \subseteq Q^r$, or even better, of type $Q^{(E (r-1)+1)} \subseteq Q^r$ for all $r>0$. The final section consolidates remarks on how often we can utilize these criteria, presenting an example.

math.AC↗

arXiv2017

Counting Roots of Polynomials over $\mathbb{Z}/p^2\mathbb{Z}$

Until recently, the only known method of finding the roots of polynomials over prime power rings, other than fields, was brute force. One reason for this is the lack of a division algorithm, obstructing the use of greatest common divisors. Fix a prime $p \in \mathbb{Z}$ and $f \in ( \mathbb{Z}/p^n \mathbb{Z} ) [x]$ any nonzero polynomial of degree $d$ whose coefficients are not all divisible by $p$. For the case $n=2$, we prove a new efficient algorithm to count the roots of $f$ in $\mathbb{Z}/p^2\mathbb{Z}$ within time polynomial in $(d+\operatorname{size}(f)+\log{p})$, and record a concise formula for the number of roots, formulated by Cheng, Gao, Rojas, and Wan.

math.NT↗

arXiv2017

Rational Singularities and Uniform Symbolic Topologies

Take $(R, \mathfrak{m})$ any normal Noetherian domain, either local or $\mathbb{N}$-graded over a field. We study the question of when $R$ satisfies the uniform symbolic topology property (USTP) of Huneke, Katz, and Validashti: namely, that there exists an integer $D>0$ such that for all prime ideals $P \subseteq R$, the symbolic power $P^{(Da)} \subseteq P^a$ for all $a >0$. Reinterpreting results of Lipman, we deduce that when $R$ is a two-dimensional rational singularity, then it satisfies the USTP. Emphasizing the non-regular setting, we produce explicit, effective multipliers $D$, working in two classes of surface singularities in equal characteristic over an algebraically closed field, using: (1) the volume of a parallelogram in $\mathbb{R}^2$ when $R$ is the coordinate ring of a simplicial toric surface; or (2) known invariants of du Val isolated singularities in characteristic zero due to Lipman.

math.AC↗

arXiv2017

Uniform Symbolic Topologies in Normal Toric Rings

Given a normal toric algebra $R$, we compute a uniform integer $D = D(R) > 0$ such that the symbolic power $P^{(D N)} \subseteq P^N$ for all $N >0$ and all monomial primes $P$. We compute the multiplier $D$ explicitly in terms of the polyhedral cone data defining $R$. In this toric setting, we draw a connection with the F-signature of $R$ in positive characteristic.

math.AC↗