**m2r** is a new R package that provides a persistent
connection between R and Macaulay2 (M2).

The package grew out of a collaboration at the 2016 Mathematics Research Community on algebraic statistics, funded by the National Science Foundation through the American Mathematical Society.

If you have a feature request, please file an issue!

**m2r** is loaded like any other R package:

```
library(m2r)
# Loading required package: mpoly
# Please cite m2r! See citation("m2r") for details.
# M2 found in /Applications/Macaulay2-1.10/bin
```

When loaded, **m2r** initializes a persistent connection
to a back-end Macaulay2 session. The basic function in R that accesses
this connection is `m2()`

, which simply accepts a character
string that is run by the Macaulay2 session.

```
m2("1 + 1")
# Starting M2... done.
# [1] "2"
```

You can see the persistence by setting variables and accessing them
across different `m2()`

calls:

```
m2("a = 1")
# [1] "1"
m2("a")
# [1] "1"
```

You can check the variables defined in the M2 session with
`m2_ls()`

:

```
m2_ls()
# [1] "a"
```

You can also check if variables exist with
`m2_exists()`

:

```
m2_exists("a")
# [1] TRUE
m2_exists(c("a","b"))
# [1] TRUE FALSE
```

Apart from the basic connection to M2, **m2r** has basic
data structures and methods to reference and manipulate the M2 objects
within R. For more on this, see the **m2r** internals
section below.

**m2r** currently has basic support for rings
(think: polynomial
rings):

```
<- ring("t", "x", "y", "z", coefring = "QQ"))
(QQtxyz # M2 Ring: QQ[t,x,y,z], grevlex order
```

and ideals of rings:

```
<- ideal("t^4 - x", "t^3 - y", "t^2 - z"))
(I # M2 Ideal of ring QQ[t,x,y,z] (grevlex) with generators :
# < t^4 - x, t^3 - y, t^2 - z >
```

You can compute Grobner bases as
well. The basic function to do this is `gb()`

:

```
gb(I)
# z^2 - x
# z t - y
# -1 z x + y^2
# -1 x + t y
# -1 z y + x t
# -1 z + t^2
```

Perhaps an easier way to do this is just to list off the polynomials as character strings:

```
gb("t^4 - x", "t^3 - y", "t^2 - z")
# z^2 - x
# z t - y
# -1 z x + y^2
# -1 x + t y
# -1 z y + x t
# -1 z + t^2
```

The result is an `mpolyList`

object, from the **mpoly**
package. You can see the M2 code by adding
`code = TRUE`

:

```
gb("t^4 - x", "t^3 - y", "t^2 - z", code = TRUE)
# m2rintgb00000003 = gb(m2rintideal00000003); gens m2rintgb00000003
```

You can compute the basis respective of different monomial orders
as well. The default ordering is the one in the respective ring, which
defaults to `grevlex`

; however, changing the order is as
simple as changing the ring.

```
ring("x", "y", "t", "z", coefring = "QQ", order = "lex")
# M2 Ring: QQ[x,y,t,z], lex order
gb("t^4 - x", "t^3 - y", "t^2 - z")
# t^2 - z
# -1 t z + y
# -1 z^2 + x
```

On a technical level, `ring()`

, `ideal()`

, and
`gb()`

use nonstandard
evaluation rules. A more stable way to use these functions is to use
their standard evaluation versions `ring_()`

,
`ideal_()`

, and `gb_()`

. Each accepts first a data
structure describing the relevant object of interest first as its own
object. For example, at a basic level this simply changes the previous
syntax to

```
use_ring(QQtxyz)
<- c("t^4 - x", "t^3 - y", "t^2 - z")
poly_chars gb_(poly_chars)
# z^2 - x
# z t - y
# -1 z x + y^2
# -1 x + t y
# -1 z y + x t
# -1 z + t^2
```

`gb_()`

is significantly easier to code with than
`gb()`

in the sense that its inputs and outputs are more
predictable, so we strongly recommend that you use `gb_()`

,
especially inside of other functions and packages.

As far as other kinds of computations are concerned, we present a potpurri of examples below.

Ideal saturation:

```
ring("x", coefring = "QQ")
# M2 Ring: QQ[x], grevlex order
<- ideal("(x-1) x (x+1)")
I saturate(I, "x") # = (x-1) (x+1)
# M2 Ideal of ring QQ[x] (grevlex) with generator :
# < x^2 - 1 >
```

Radicalization:

```
<- ideal("x^2")
I radical(I)
# M2 Ideal of ring QQ[x] (grevlex) with generator :
# < x >
```

Primary decomposition:

```
ring("x", "y", "z", coefring = "QQ")
# M2 Ring: QQ[x,y,z], grevlex order
<- ideal("x z", "y z")
I primary_decomposition(I)
# M2 List of ideals of QQ[x,y,z] (grevlex) :
# < z >
# < x, y >
```

Dimension:

```
ring("x", "y", coefring = "QQ")
# M2 Ring: QQ[x,y], grevlex order
<- ideal("y - (x+1)")
I dimension(I)
# [1] 1
```

You can compute prime
decompositions of integers with `factor_n()`

:

```
<- 2^5 * 3^4 * 5^3 * 7^2 * 11^1)
(x # [1] 174636000
factor_n(x)
# $prime
# [1] 2 3 5 7 11
#
# $power
# [1] 5 4 3 2 1
```

You can also factor
polynomials over rings using `factor_poly()`

:

```
factor_poly("x^4 - y^4")
# $factor
# x - y
# x + y
# x^2 + y^2
#
# $power
# [1] 1 1 1
```

The Smith normal form of a matrix *M* here refers to the
decomposition of an integer matrix *D = PMQ*, where *D*,
*P*, and *Q* are integer matrices and *D* is
diagonal. *P* and *Q* are unimodular matrices (their
determinants are -1 or 1), so they are invertible. This is somewhat like
a singular value decomposition for integer matrices.

```
<- matrix(c(
M 2, 4, 4,
-6, 6, 12,
10, -4, -16
nrow = 3, byrow = TRUE)
),
<- snf(M))
(mats # $D
# [,1] [,2] [,3]
# [1,] 12 0 0
# [2,] 0 6 0
# [3,] 0 0 2
# M2 Matrix over ZZ[]
# $P
# [,1] [,2] [,3]
# [1,] 1 0 1
# [2,] 0 1 0
# [3,] 0 0 1
# M2 Matrix over ZZ[]
# $Q
# [,1] [,2] [,3]
# [1,] 4 -2 -1
# [2,] -2 3 1
# [3,] 3 -2 -1
# M2 Matrix over ZZ[]
<- mats$P; D <- mats$D; Q <- mats$Q
P
%*% M %*% Q # = D
P # [,1] [,2] [,3]
# [1,] 12 0 0
# [2,] 0 6 0
# [3,] 0 0 2
solve(P) %*% D %*% solve(Q) # = M
# [,1] [,2] [,3]
# [1,] 2 4 4
# [2,] -6 6 12
# [3,] 10 -4 -16
det(P)
# [1] 1
det(Q)
# [1] -1
```

`m2`

objectsAt a basic level, **m2r** works by passing strings
between R and M2. Originating at the R side, these strings are properly
formated M2 code constructed from the inputs to the R functions. That
code goes to M2, is evaluated there, and then “exported” with M2’s
function `toExternalString()`

. The resulting string often,
but not always, produces the M2 code needed to recreate the object
resulting from the evaluation, and in that sense is M2’s version of R’s
`dput()`

. That string is passed back into R and parsed there
into R-style data structures, typically S3-classed
lists.

The R-side parsing of the external string from M2 is an expensive
process because it is currently implemented in R. Consequently (and for
other reasons, too!), in some cases you’ll want to do a M2 computation
from R, but leave the output in M2. Since you will ultimately want
something in R referencing the result, nearly every **m2r**
function that performs M2 computations has a pointer version. As a
simple naming convention, the name of the function that returns the
pointer, called the reference function, is determined by the name of the
ordinary function, called the value function, by appending a
`.`

.

For example, we’ve seen that `factor_n()`

computes the
prime decomposition of a number. The corresponding reference function is
`factor_n.()`

:

```
<- 2^5 * 3^4 * 5^3 * 7^2 * 11^1)
(x # [1] 174636000
factor_n.(x)
# M2 Pointer Object
# ExternalString : new Product from {new Power from {2,5},new Power fro...
# M2 Name : m2o460
# M2 Class : Product (WrapperType)
```

All value functions simply wrap reference functions and parse the
output with `m2_parse()`

, a general M2 parser, often followed
by a little more parsing. `m2_parse()`

typically creates an
object of class `m2`

so that R knows what kind of thing it
is. For example:

```
class(factor_n.(x))
# [1] "m2_pointer" "m2"
```

Even more, `m2_parse()`

often creates objects that have an
inheritance structure that references `m2`

somewhere in the
middle of its class structure, with specific structure preceding and
general structure succeeding (examples below). Apart from its class, the
general principle we follow here for the object itself is this: if the
M2 object has a direct analogue in R, it is parsed into that kind of R
object and additional M2 properties are kept as metadata (attributes);
if there is no direct analogue in R, the object is an `NA`

with metadata.

Perhaps the easiest way to see this is with a matrix.
`m2_matrix()`

creates a matrix on the M2 side from input on
the R side. In the following, to make things more clear we use **magrittr**’s
pipe operator, with which the following calls are semantically
equivalent: `g(f(x))`

and
`x %>% f %>% g`

.

```
library(magrittr)
<- matrix(c(1,2,3,4,5,6), nrow = 3, ncol = 2)
mat %>% m2_matrix. # = m2_matrix.(mat)
mat # M2 Pointer Object
# ExternalString : map((ZZ)^3,(ZZ)^2,{{1, 4}, {2, 5}, {3, 6}})
# M2 Name : m2rintmatrix00000001
# M2 Class : Matrix (Type)
%>% m2_matrix. %>% m2_parse
mat # [,1] [,2]
# [1,] 1 4
# [2,] 2 5
# [3,] 3 6
# M2 Matrix over ZZ[]
%>% m2_matrix. %>% m2_parse %>% str
mat # 'm2_matrix' int [1:3, 1:2] 1 2 3 4 5 6
# - attr(*, "m2_name")= chr "m2rintmatrix00000003"
# - attr(*, "m2_meta")=List of 1
# ..$ ring: 'm2_polynomialring' logi NA
# .. ..- attr(*, "m2_name")= chr "ZZ"
# .. ..- attr(*, "m2_meta")=List of 3
# .. .. ..$ vars : NULL
# .. .. ..$ coefring: chr "ZZ"
# .. .. ..$ order : chr "grevlex"
%>% m2_matrix # = m2_parse(m2_matrix.(mat))
mat # [,1] [,2]
# [1,] 1 4
# [2,] 2 5
# [3,] 3 6
# M2 Matrix over ZZ[]
```

It may be helpful to think of every `m2`

object as being a
missing value (`NA`

, a `logical(1)`

) with two M2
attributes: their name (`m2_name`

) and a capture-all named
list (`m2_meta`

). These can be accessed with
`m2_name()`

and `m2_meta()`

. For example, a ring,
having no analogous object in R, is an `NA`

with
attributes:

```
<- ring("x", "y", coefring = "QQ")
r str(r)
# 'm2_polynomialring' logi NA
# - attr(*, "m2_name")= chr "m2rintring00000006"
# - attr(*, "m2_meta")=List of 3
# ..$ vars :List of 2
# .. ..$ : chr "x"
# .. ..$ : chr "y"
# ..$ coefring: chr "QQ"
# ..$ order : chr "grevlex"
class(r)
# [1] "m2_polynomialring" "m2"
m2_name(r)
# [1] "m2rintring00000006"
m2_meta(r)
# $vars
# $vars[[1]]
# [1] "x"
#
# $vars[[2]]
# [1] "y"
#
#
# $coefring
# [1] "QQ"
#
# $order
# [1] "grevlex"
```

But a matrix of integers isn’t:

```
<- m2_matrix(matrix(c(1,2,3,4,5,6), nrow = 3, ncol = 2))
mat str(mat)
# 'm2_matrix' num [1:3, 1:2] 1 2 3 4 5 6
# - attr(*, "m2_name")= chr "m2rintmatrix00000005"
# - attr(*, "m2_meta")=List of 1
# ..$ ring: 'm2_polynomialring' logi NA
# .. ..- attr(*, "m2_name")= chr "ZZ"
# .. ..- attr(*, "m2_meta")=List of 3
# .. .. ..$ vars : NULL
# .. .. ..$ coefring: chr "ZZ"
# .. .. ..$ order : chr "grevlex"
class(mat)
# [1] "m2_matrix" "m2" "matrix"
m2_name(mat)
# [1] "m2rintmatrix00000005"
m2_meta(mat)
# $ring
# M2 Ring: ZZ[], grevlex order
```

Since a matrix of integers is an object in R, it’s represented as
one, and consequently we can compute with it directly as it if it were a
matrix; it is. On the other hand, since a ring is not, it’s an
`NA`

. When dealing with M2, object like rings, that is to say
objects without R analogues, are more common than those like integer
matrices.

We’ve already wrapped a number of Macaulay2 functions; for a list of
functions in **m2r**, check out
`ls("package:m2r")`

. But the list is very far from
exhaustive. To create your own wrapper function of a Macaulay2 command,
you’ll need to create an R file that looks like the one below. This will
create both value (e.g. `f`

) and reference/pointer
(e.g. `f.`

) versions of the function. As a good example of
these at work, see the scripts for `factor_n()`

or `factor_poly()`

.

```
#' Function documentation header
#'
#' Function header explanation, can run several lines. Function
#' header explanation, can run several lines. Function header
#' explanation, can run several lines.
#'
#' @param esntl_parm_1 esntl_parm_1 description
#' @param esntl_parm_2 esntl_parm_2 description
#' @param code return only the M2 code? (default: \code{FALSE})
#' @param parse_parm_1 parse_parm_1 description
#' @param parse_parm_2 parse_parm_2 description
#' @param ... ...
#' @name f
#' @return (value version) parsed output or (reference/dot version)
#' \code{m2_pointer}
#' @examples
#'
#' \dontrun{ requires Macaulay2 be installed
#'
#' # put examples here
#' 1 + 1
#'
#' }
#'
# value version of f (standard user version)
#' @rdname f
#' @export
<- function(esntl_parm_1, esntl_parm_2, code = FALSE, parse_parm_1, parse_parm_2, ...) {
f
# run m2
<- as.list(match.call())[-1]
args <- lapply(args, eval, envir = parent.frame())
eargs <- do.call(f., eargs)
pointer if(code) return(invisible(pointer))
# parse output
<- m2_parse(pointer)
parsed_out
# more parsing, like changing classes and such
TRUE
# return
TRUE
}
# reference version of f (returns pointer to m2 object)
#' @rdname f
#' @export
<- function(esntl_parm_1, esntl_parm_2, code = FALSE, ...) {
f.
# basic arg checking
TRUE
# create essential parameters to pass to m2 this step regularizes input to m2, so it
# is the one that deals with pointers, chars, rings, ideals, mpolyLists, etc.
TRUE
# construct m2_code from regularized essential parameters
TRUE
# message
if(code) { message(m2_code); return(invisible(m2_code)) }
# run m2 and return pointer
m2.(m2_code)
}
```

This material is based upon work supported by the National Science Foundation under Grant Nos. 1321794 and 1622449.

Here’s how you can install the current *developmental* version
of **m2r**. Remember you need to have Macaulay2 downloaded;
**m2r** will look for it in your path variable (in the
terminal, `echo $PATH`

) as set by
`~/.bash_profile`

or, if nonexistent, then
`~/.bashrc`

, then `~/.profile`

.

```
# install.packages("devtools")
::install_github("coneill-math/m2r") devtools
```