This package gives the ability to compute symbolic powers, and related invariants, of ideals in a polynomial ring or a quotient of a polynomial ring. For example, in the context of the default behavior, symbolicPower assumes the following definition of the symbolic power of an ideal *I*,

as defined by M. Hochster and C. Huneke.

Alternatively, as defined in Villarreal, symbolicPower has the option to restrict to minimal primes versus use all associated primes with UseMinimalPrimes. In particular, the symbolic power of an ideal *I* is defined as

where *Min(R/I)* is the set of minimal primes in *I*,

- M. Hochster and C. Huneke.
*Comparison of symbolic and ordinary powers of ideals.*Invent. Math. 147 (2002), no. 2, 349–369. - R. Villarreal.
*Monomial algebras.*Second edition. Monographs and Research Notes in Mathematics. CRC Press, Boca Raton, FL, 2015. xviii+686 pp. ISBN: 978-1-4822-3469-5. - Hailong Dao, Alessandro De Stefani, Eloísa Grifo, Craig Huneke, and Luis Núñez-Betancourt.
*Symbolic powers of ideals.*in Singularities and foliations. Geometry, topology and applications, pp. 387-432, Springer Proc. Math. Stat., 222, Springer, Cham, 2018. See https://arxiv.org/abs/1708.03010.

- Ben Drabkin
- Andrew Conner
- Alexandra Seceleanu
- Branden Stone
- Xuehua (Diana) Zhong

- Computing symbolic powers of an ideal
- Alternative algorithm to compute the symbolic powers of a prime ideal in positive characteristic

Version **2.0** of this package was accepted for publication in volume 9 of the journal The Journal of Software for Algebra and Geometry on 20 May 2019, in the article Calculations involving symbolic powers. That version can be obtained from the journal or from the *Macaulay2* source code repository, http://github.com/Macaulay2/M2/blob/master/M2/Macaulay2/packages/SymbolicPowers.m2, commit number fe3eea250b0c2c9a0ebbbd84cf44b7a52da63fc0.

- Functions and commands
- assPrimesHeight -- The heights of all associated primes
- asymptoticRegularity -- approximates the asymptotic regularity
- bigHeight -- computes the big height of an ideal
- containmentProblem -- computes the smallest symbolic power contained in a power of an ideal.
- isKonig -- determines if a given square-free ideal is Konig.
- isPacked -- determines if a given square-free ideal is packed.
- isSymbolicEqualOrdinary -- tests if symbolic power is equal to ordinary power
- isSymbPowerContainedinPower -- tests if the m-th symbolic power an ideal is contained the n-th power
- joinIdeals -- Computes the join of the given ideals
- lowerBoundResurgence -- computes a lower bound for the resurgence of a given ideal.
- minDegreeSymbPower -- returns the minimal degree of a given symbolic power of an ideal.
- minimalPart -- intersection of the minimal components
- noPackedAllSubs -- finds all substitutions of variables by 1 and/or 0 for which ideal is not Konig.
- noPackedSub -- finds a substitution of variables by 1 and/or 0 for which an ideal is not Konig.
- squarefreeGens -- returns all square-free monomials in a minimal generating set of the given ideal.
- squarefreeInCodim -- finds square-fee monomials in ideal raised to the power of the codimension.
- symbolicDefect -- computes the symbolic defect of an ideal
- symbolicPolyhedron -- computes the symbolic polyhedron for a monomial ideal.
- symbolicPower -- computes the symbolic power of an ideal.
- symbolicPowerJoin -- computes the symbolic power of the prime ideal using join of ideals.
- symbPowerPrimePosChar
- waldschmidt -- computes the Waldschmidt constant for a homogeneous ideal.

- Symbols
- CIPrimes -- an option to compute the symbolic power by taking the intersection of the powers of the primary components
- InSymbolic -- an optional parameter used in containmentProblem.
- SampleSize -- optional parameter used for approximating asymptotic invariants that are defined as limits.
- UseMinimalPrimes -- an option to only use minimal primes to calculate symbolic powers
- UseWaldschmidt -- optional input for computing a lower bound for the resurgence of a given ideal.