Using Macaulay2 from within R : the m2r package Christopher ONeill - - PowerPoint PPT Presentation

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Using Macaulay2 from within R : the m2r package Christopher ONeill - - PowerPoint PPT Presentation

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Using Macaulay2 from within R : the m2r package Christopher ONeill University of California Davis coneill@math.ucdavis.edu Joint with David Kahle and Jeff Sommars Mathematics Research Communities on Algebraic Statistics August 3, 2017

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Using Macaulay2 from within R: the m2r package

Christopher O’Neill

University of California Davis coneill@math.ucdavis.edu Joint with David Kahle and Jeff Sommars Mathematics Research Communities on Algebraic Statistics

August 3, 2017

Christopher O’Neill (UC Davis) The m2r package August 3, 2017 1 / 20

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R and Macaulay2

R: a statistician’s best friend Data storage and manipulation, array calculations, data analysis, . . .

Christopher O’Neill (UC Davis) The m2r package August 3, 2017 2 / 20

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R and Macaulay2

R: a statistician’s best friend Data storage and manipulation, array calculations, data analysis, . . . Macaulay2: an algebraic geometer’s best friend Polynomial ideals, Gr¨

  • bner bases, Hilbert functions, . . .

Christopher O’Neill (UC Davis) The m2r package August 3, 2017 2 / 20

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R and Macaulay2

R: a statistician’s best friend Data storage and manipulation, array calculations, data analysis, . . . Macaulay2: an algebraic geometer’s best friend Polynomial ideals, Gr¨

  • bner bases, Hilbert functions, . . .

Algebraic statisticians: best of both worlds

Christopher O’Neill (UC Davis) The m2r package August 3, 2017 2 / 20

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Running Macaulay2 from R the old way

R version 3.3.0 ... >

Christopher O’Neill (UC Davis) The m2r package August 3, 2017 3 / 20

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Running Macaulay2 from R the old way

R version 3.3.0 ... > library("algstat") >

Christopher O’Neill (UC Davis) The m2r package August 3, 2017 3 / 20

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Running Macaulay2 from R the old way

R version 3.3.0 ... > library("algstat") > code <- "R = QQ[x,y,z] I = ideal(xˆ2, x*y, xˆ3*yˆ2) gens gb I" >

Christopher O’Neill (UC Davis) The m2r package August 3, 2017 3 / 20

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Running Macaulay2 from R the old way

R version 3.3.0 ... > library("algstat") > code <- "R = QQ[x,y,z] I = ideal(xˆ2, x*y, xˆ3*yˆ2) gens gb I" > m2(code) [1] "R" [1] "ideal(xˆ2,x*y,xˆ3*yˆ2)" [1] "matrix {{x*y, xˆ2}}"

Christopher O’Neill (UC Davis) The m2r package August 3, 2017 3 / 20

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Running Macaulay2 from R the old way

R version 3.3.0 ... > library("algstat") > code <- "R = QQ[x,y,z] I = ideal(xˆ2, x*y, xˆ3*yˆ2) gens gb I" > m2(code) [1] "R" [1] "ideal(xˆ2,x*y,xˆ3*yˆ2)" [1] "matrix {{x*y, xˆ2}}"

m2Code.m2

f = "m2Out" << "" f << toString( R = QQ[x,y,z] ) << endl f << toString( I = ideal(xˆ2, x*y, xˆ3*yˆ2) ) << endl f << toString( gens gb I ) << endl f << close

Christopher O’Neill (UC Davis) The m2r package August 3, 2017 3 / 20

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The m2r package in action

R version 3.3.0 ... >

Christopher O’Neill (UC Davis) The m2r package August 3, 2017 4 / 20

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The m2r package in action

R version 3.3.0 ... > library("m2r") Loading required package: mpoly Loading required package: stringr please cite mpoly if you use it; see citation("mpoly") M2 found in /usr/local/macaulay2/bin >

Christopher O’Neill (UC Davis) The m2r package August 3, 2017 4 / 20

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The m2r package in action

R version 3.3.0 ... > library("m2r") Loading required package: mpoly Loading required package: stringr please cite mpoly if you use it; see citation("mpoly") M2 found in /usr/local/macaulay2/bin > start_m2() Starting M2... done. >

Christopher O’Neill (UC Davis) The m2r package August 3, 2017 4 / 20

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The m2r package in action

R version 3.3.0 ... > library("m2r") Loading required package: mpoly Loading required package: stringr please cite mpoly if you use it; see citation("mpoly") M2 found in /usr/local/macaulay2/bin > start_m2() Starting M2... done. > m2("1+1") [1] "2" >

Christopher O’Neill (UC Davis) The m2r package August 3, 2017 4 / 20

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The m2r package in action

R version 3.3.0 ... > library("m2r") Loading required package: mpoly Loading required package: stringr please cite mpoly if you use it; see citation("mpoly") M2 found in /usr/local/macaulay2/bin > start_m2() Starting M2... done. > m2("1+1") [1] "2" > m2("a = 5") [1] "5" >

Christopher O’Neill (UC Davis) The m2r package August 3, 2017 4 / 20

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The m2r package in action

R version 3.3.0 ... > library("m2r") Loading required package: mpoly Loading required package: stringr please cite mpoly if you use it; see citation("mpoly") M2 found in /usr/local/macaulay2/bin > start_m2() Starting M2... done. > m2("1+1") [1] "2" > m2("a = 5") [1] "5" > m2("a") [1] "5"

Christopher O’Neill (UC Davis) The m2r package August 3, 2017 4 / 20

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Under the hood: sockets

Christopher O’Neill (UC Davis) The m2r package August 3, 2017 5 / 20

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Under the hood: sockets

m2_start()

Christopher O’Neill (UC Davis) The m2r package August 3, 2017 5 / 20

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Under the hood: sockets

m2_start() launch M2 process

Christopher O’Neill (UC Davis) The m2r package August 3, 2017 5 / 20

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Under the hood: sockets

m2_start() launch M2 process wait for available connection

Christopher O’Neill (UC Davis) The m2r package August 3, 2017 5 / 20

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Under the hood: sockets

m2_start() launch M2 process wait for available connection create server socket

Christopher O’Neill (UC Davis) The m2r package August 3, 2017 5 / 20

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Under the hood: sockets

m2_start() launch M2 process wait for available connection create server socket wait for client connection

Christopher O’Neill (UC Davis) The m2r package August 3, 2017 5 / 20

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Under the hood: sockets

m2_start() launch M2 process wait for available connection connect to socket create server socket wait for client connection

Christopher O’Neill (UC Davis) The m2r package August 3, 2017 5 / 20

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Under the hood: sockets

m2_start() launch M2 process wait for available connection connect to socket create server socket wait for client connection

Christopher O’Neill (UC Davis) The m2r package August 3, 2017 5 / 20

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Under the hood: sockets

m2_start() launch M2 process wait for available connection connect to socket wait for message from server create server socket wait for client connection

Christopher O’Neill (UC Davis) The m2r package August 3, 2017 5 / 20

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Under the hood: sockets

m2_start() launch M2 process wait for available connection connect to socket wait for message from server create server socket wait for client connection send "1.0.0"

Christopher O’Neill (UC Davis) The m2r package August 3, 2017 5 / 20

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Under the hood: sockets

m2_start() launch M2 process wait for available connection connect to socket wait for message from server create server socket wait for client connection send "1.0.0" wait for input from client

Christopher O’Neill (UC Davis) The m2r package August 3, 2017 5 / 20

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Under the hood: sockets

m2_start() launch M2 process wait for available connection connect to socket wait for message from server receive "1.0.0" create server socket wait for client connection send "1.0.0" wait for input from client

Christopher O’Neill (UC Davis) The m2r package August 3, 2017 5 / 20

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Under the hood: sockets

m2_start() launch M2 process wait for available connection connect to socket wait for message from server receive "1.0.0" verify version match create server socket wait for client connection send "1.0.0" wait for input from client

Christopher O’Neill (UC Davis) The m2r package August 3, 2017 5 / 20

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Under the hood: sockets

m2_start() launch M2 process wait for available connection connect to socket wait for message from server receive "1.0.0" verify version match return from m2_start() create server socket wait for client connection send "1.0.0" wait for input from client

Christopher O’Neill (UC Davis) The m2r package August 3, 2017 5 / 20

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Under the hood: sockets

wait for input from client

Christopher O’Neill (UC Davis) The m2r package August 3, 2017 6 / 20

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Under the hood: sockets

m2("1+1") wait for input from client

Christopher O’Neill (UC Davis) The m2r package August 3, 2017 6 / 20

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Under the hood: sockets

m2("1+1") send "1+1" to server wait for input from client

Christopher O’Neill (UC Davis) The m2r package August 3, 2017 6 / 20

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Under the hood: sockets

m2("1+1") send "1+1" to server wait for response wait for input from client

Christopher O’Neill (UC Davis) The m2r package August 3, 2017 6 / 20

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Under the hood: sockets

m2("1+1") send "1+1" to server wait for response wait for input from client receive "1+1" from client

Christopher O’Neill (UC Davis) The m2r package August 3, 2017 6 / 20

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Under the hood: sockets

m2("1+1") send "1+1" to server wait for response wait for input from client receive "1+1" from client evaluate "1+1" to "2"

Christopher O’Neill (UC Davis) The m2r package August 3, 2017 6 / 20

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Under the hood: sockets

m2("1+1") send "1+1" to server wait for response wait for input from client receive "1+1" from client evaluate "1+1" to "2" send "2" to client

Christopher O’Neill (UC Davis) The m2r package August 3, 2017 6 / 20

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Under the hood: sockets

m2("1+1") send "1+1" to server wait for response wait for input from client receive "1+1" from client evaluate "1+1" to "2" send "2" to client wait for input from client

Christopher O’Neill (UC Davis) The m2r package August 3, 2017 6 / 20

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Under the hood: sockets

m2("1+1") send "1+1" to server wait for response receive "2" from server wait for input from client receive "1+1" from client evaluate "1+1" to "2" send "2" to client wait for input from client

Christopher O’Neill (UC Davis) The m2r package August 3, 2017 6 / 20

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Under the hood: sockets

m2("1+1") send "1+1" to server wait for response receive "2" from server return "2" from m2() wait for input from client receive "1+1" from client evaluate "1+1" to "2" send "2" to client wait for input from client

Christopher O’Neill (UC Davis) The m2r package August 3, 2017 6 / 20

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Under the hood: sockets

Christopher O’Neill (UC Davis) The m2r package August 3, 2017 7 / 20

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Under the hood: sockets

>

Christopher O’Neill (UC Davis) The m2r package August 3, 2017 7 / 20

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Under the hood: sockets

> m2("a = 5") [1] "5" > m2("a") [1] "5" >

Christopher O’Neill (UC Davis) The m2r package August 3, 2017 7 / 20

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Under the hood: sockets

> m2("a = 5") [1] "5" > m2("a") [1] "5" > m2("1+") Error: Macaulay2 Error! >

Christopher O’Neill (UC Davis) The m2r package August 3, 2017 7 / 20

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Under the hood: sockets

> m2("a = 5") [1] "5" > m2("a") [1] "5" > m2("1+") Error: Macaulay2 Error! > m2("2+1") [1] "3"

Christopher O’Neill (UC Davis) The m2r package August 3, 2017 7 / 20

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Under the hood: sockets

>

Christopher O’Neill (UC Davis) The m2r package August 3, 2017 8 / 20

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Under the hood: sockets

> start_m2() Starting M2... done. > m2("1+1") [1] "2" >

Christopher O’Neill (UC Davis) The m2r package August 3, 2017 8 / 20

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Under the hood: sockets

> start_m2() Starting M2... done. > m2("1+1") [1] "2" >

  • So. . . now what?

Christopher O’Neill (UC Davis) The m2r package August 3, 2017 8 / 20

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Under the hood: sockets

> start_m2() Starting M2... done. > m2("1+1") [1] "2" >

  • So. . . now what?

New features since creation: Lots of convenience functions High-level parser Cloud computing

Christopher O’Neill (UC Davis) The m2r package August 3, 2017 8 / 20

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Convenience functions

> m2("a = 5") [1] "5" > m2("a") [1] "5" >

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Convenience functions

> m2("a = 5") [1] "5" > m2("a") [1] "5" > m2("R = QQ[x,y,z]") [1] "QQ(monoid[x..z, Degrees => {3:1}, Heft => {1}, MonomialOrder => VerticalList{MonomialSize => 32, GRevLex => {3:1}, Position => Up}, DegreeRank => 1])" >

Christopher O’Neill (UC Davis) The m2r package August 3, 2017 9 / 20

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Convenience functions

> m2("a = 5") [1] "5" > m2("a") [1] "5" > m2("R = QQ[x,y,z]") [1] "QQ(monoid[x..z, Degrees => {3:1}, Heft => {1}, MonomialOrder => VerticalList{MonomialSize => 32, GRevLex => {3:1}, Position => Up}, DegreeRank => 1])" > m2("I = ideal(xˆ2, x*y, xˆ3*yˆ2)") [1] "ideal map((R)ˆ1,(R)ˆ{{-2},{-2},{-5}},{{xˆ2, x*y, xˆ3*yˆ2}})" >

Christopher O’Neill (UC Davis) The m2r package August 3, 2017 9 / 20

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Convenience functions

> m2("a = 5") [1] "5" > m2("a") [1] "5" > m2("R = QQ[x,y,z]") [1] "QQ(monoid[x..z, Degrees => {3:1}, Heft => {1}, MonomialOrder => VerticalList{MonomialSize => 32, GRevLex => {3:1}, Position => Up}, DegreeRank => 1])" > m2("I = ideal(xˆ2, x*y, xˆ3*yˆ2)") [1] "ideal map((R)ˆ1,(R)ˆ{{-2},{-2},{-5}},{{xˆ2, x*y, xˆ3*yˆ2}})" > m2("gens gb I") [1] "map((R)ˆ1,(R)ˆ{{-2},{-2}},{{x*y, xˆ2}})"

Christopher O’Neill (UC Davis) The m2r package August 3, 2017 9 / 20

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Convenience functions

> m2("a = 5") [1] "5" > m2("a") [1] "5" > m2("R = QQ[x,y,z]") [1] "QQ(monoid[x..z, Degrees => {3:1}, Heft => {1}, MonomialOrder => VerticalList{MonomialSize => 32, GRevLex => {3:1}, Position => Up}, DegreeRank => 1])" > m2("I = ideal(xˆ2, x*y, xˆ3*yˆ2)") [1] "ideal map((R)ˆ1,(R)ˆ{{-2},{-2},{-5}},{{xˆ2, x*y, xˆ3*yˆ2}})" > m2("gens gb I") [1] "map((R)ˆ1,(R)ˆ{{-2},{-2}},{{x*y, xˆ2}})"

Christopher O’Neill (UC Davis) The m2r package August 3, 2017 9 / 20

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Convenience functions

> m2("a = 5") [1] "5" > m2("a") [1] "5" >

Christopher O’Neill (UC Davis) The m2r package August 3, 2017 10 / 20

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Convenience functions

> m2("a = 5") [1] "5" > m2("a") [1] "5" > (R <- ring("x", "y", "z", coefring = "QQ")) M2 Ring: QQ[x,y,z], grevlex order >

Christopher O’Neill (UC Davis) The m2r package August 3, 2017 10 / 20

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Convenience functions

> m2("a = 5") [1] "5" > m2("a") [1] "5" > (R <- ring("x", "y", "z", coefring = "QQ")) M2 Ring: QQ[x,y,z], grevlex order > (I <- ideal("xˆ2", "x*y", "xˆ3*yˆ2")) M2 Ideal of ring QQ[x,y,z] (grevlex) with generators : < xˆ2, x y, xˆ3 yˆ2 > >

Christopher O’Neill (UC Davis) The m2r package August 3, 2017 10 / 20

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Convenience functions

> m2("a = 5") [1] "5" > m2("a") [1] "5" > (R <- ring("x", "y", "z", coefring = "QQ")) M2 Ring: QQ[x,y,z], grevlex order > (I <- ideal("xˆ2", "x*y", "xˆ3*yˆ2")) M2 Ideal of ring QQ[x,y,z] (grevlex) with generators : < xˆ2, x y, xˆ3 yˆ2 > > (mygens <- gb(I)) x y xˆ2 >

Christopher O’Neill (UC Davis) The m2r package August 3, 2017 10 / 20

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Convenience functions

> m2("a = 5") [1] "5" > m2("a") [1] "5" > (R <- ring("x", "y", "z", coefring = "QQ")) M2 Ring: QQ[x,y,z], grevlex order > (I <- ideal("xˆ2", "x*y", "xˆ3*yˆ2")) M2 Ideal of ring QQ[x,y,z] (grevlex) with generators : < xˆ2, x y, xˆ3 yˆ2 > > (mygens <- gb(I)) x y xˆ2 > mygens[[2]] xˆ2

Christopher O’Neill (UC Davis) The m2r package August 3, 2017 10 / 20

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Convenience functions

> m2("a = 5") [1] "5" > m2("a") [1] "5" > (R <- ring("x", "y", "z", coefring = "QQ")) M2 Ring: QQ[x,y,z], grevlex order > (I <- ideal("xˆ2", "x*y", "xˆ3*yˆ2")) M2 Ideal of ring QQ[x,y,z] (grevlex) with generators : < xˆ2, x y, xˆ3 yˆ2 > > (mygens <- gb(I)) x y xˆ2 > mygens[[2]] ← − mpoly xˆ2

Christopher O’Neill (UC Davis) The m2r package August 3, 2017 10 / 20

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Convenience functions

> m2("a = 5") [1] "5" > m2("a") [1] "5" > (R <- ring("x", "y", "z", coefring = "QQ")) M2 Ring: QQ[x,y,z], grevlex order > (I <- ideal("xˆ2", "x*y", "xˆ3*yˆ2")) M2 Ideal of ring QQ[x,y,z] (grevlex) with generators : < xˆ2, x y, xˆ3 yˆ2 > > (mygens <- gb(I)) ← − mpolylist x y xˆ2 > mygens[[2]] ← − mpoly xˆ2

Christopher O’Neill (UC Davis) The m2r package August 3, 2017 10 / 20

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Convenience functions

> (I <- ideal("xˆ2", "x*y", "xˆ3*yˆ2")) M2 Ideal of ring QQ[x,y,z] (grevlex) with generators : < xˆ2, x y, xˆ3 yˆ2 > >

Christopher O’Neill (UC Davis) The m2r package August 3, 2017 11 / 20

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Convenience functions

> (I <- ideal("xˆ2", "x*y", "xˆ3*yˆ2")) M2 Ideal of ring QQ[x,y,z] (grevlex) with generators : < xˆ2, x y, xˆ3 yˆ2 > > radical(I) M2 Ideal of ring QQ[x,y,z] (grevlex) with generator : < x > >

Christopher O’Neill (UC Davis) The m2r package August 3, 2017 11 / 20

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Convenience functions

> (I <- ideal("xˆ2", "x*y", "xˆ3*yˆ2")) M2 Ideal of ring QQ[x,y,z] (grevlex) with generators : < xˆ2, x y, xˆ3 yˆ2 > > radical(I) M2 Ideal of ring QQ[x,y,z] (grevlex) with generator : < x > > saturate(I,ideal("xˆ5")) M2 Ideal of ring QQ[x,y,z] (grevlex) with generator : < 1 > >

Christopher O’Neill (UC Davis) The m2r package August 3, 2017 11 / 20

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Convenience functions

> (I <- ideal("xˆ2", "x*y", "xˆ3*yˆ2")) M2 Ideal of ring QQ[x,y,z] (grevlex) with generators : < xˆ2, x y, xˆ3 yˆ2 > > radical(I) M2 Ideal of ring QQ[x,y,z] (grevlex) with generator : < x > > saturate(I,ideal("xˆ5")) M2 Ideal of ring QQ[x,y,z] (grevlex) with generator : < 1 > > I+I M2 Ideal of ring QQ[x,y,z] (grevlex) with generators : < xˆ2, x y, xˆ3 yˆ2, xˆ2, x y, xˆ3 yˆ2 > >

Christopher O’Neill (UC Davis) The m2r package August 3, 2017 11 / 20

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Convenience functions

> (I <- ideal("xˆ2", "x*y", "xˆ3*yˆ2")) M2 Ideal of ring QQ[x,y,z] (grevlex) with generators : < xˆ2, x y, xˆ3 yˆ2 > > radical(I) M2 Ideal of ring QQ[x,y,z] (grevlex) with generator : < x > > saturate(I,ideal("xˆ5")) M2 Ideal of ring QQ[x,y,z] (grevlex) with generator : < 1 > > I+I M2 Ideal of ring QQ[x,y,z] (grevlex) with generators : < xˆ2, x y, xˆ3 yˆ2, xˆ2, x y, xˆ3 yˆ2 > > gb(I+I) x y xˆ2

Christopher O’Neill (UC Davis) The m2r package August 3, 2017 11 / 20

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Convenience functions

> (I <- ideal("xˆ2", "x*y", "xˆ3*yˆ2")) M2 Ideal of ring QQ[x,y,z] (grevlex) with generators : < xˆ2, x y, xˆ3 yˆ2 > >

Christopher O’Neill (UC Davis) The m2r package August 3, 2017 12 / 20

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Convenience functions

> (I <- ideal("xˆ2", "x*y", "xˆ3*yˆ2")) M2 Ideal of ring QQ[x,y,z] (grevlex) with generators : < xˆ2, x y, xˆ3 yˆ2 > > primary_decomposition(I) M2 List of ideals of QQ[x,y,z] (grevlex) : < x > < xˆ2, y > >

Christopher O’Neill (UC Davis) The m2r package August 3, 2017 12 / 20

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Convenience functions

> (I <- ideal("xˆ2", "x*y", "xˆ3*yˆ2")) M2 Ideal of ring QQ[x,y,z] (grevlex) with generators : < xˆ2, x y, xˆ3 yˆ2 > > primary_decomposition(I) M2 List of ideals of QQ[x,y,z] (grevlex) : < x > < xˆ2, y > > dimension(I) [1] 2 >

Christopher O’Neill (UC Davis) The m2r package August 3, 2017 12 / 20

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Convenience functions

> (I <- ideal("xˆ2", "x*y", "xˆ3*yˆ2")) M2 Ideal of ring QQ[x,y,z] (grevlex) with generators : < xˆ2, x y, xˆ3 yˆ2 > > primary_decomposition(I) M2 List of ideals of QQ[x,y,z] (grevlex) : < x > < xˆ2, y > > dimension(I) [1] 2 > ring("x", "y", "z", coefring = "QQ", code = TRUE) m2rintring00000002 = QQ[x,y,z,MonomialOrder=>{GRevLex=>3}] >

Christopher O’Neill (UC Davis) The m2r package August 3, 2017 12 / 20

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Convenience functions

> (I <- ideal("xˆ2", "x*y", "xˆ3*yˆ2")) M2 Ideal of ring QQ[x,y,z] (grevlex) with generators : < xˆ2, x y, xˆ3 yˆ2 > > primary_decomposition(I) M2 List of ideals of QQ[x,y,z] (grevlex) : < x > < xˆ2, y > > dimension(I) [1] 2 > ring("x", "y", "z", coefring = "QQ", code = TRUE) m2rintring00000002 = QQ[x,y,z,MonomialOrder=>{GRevLex=>3}] > dimension(I, code = TRUE) dim(m2rintideal00000001)

Christopher O’Neill (UC Davis) The m2r package August 3, 2017 12 / 20

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The parser

>

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The parser

> m2_matrix(matrix(c(1,2,3,4), nrow = 2, ncol = 2)) [,1] [,2] [1,] 1 3 [2,] 2 4 M2 Matrix over ZZ[] >

Christopher O’Neill (UC Davis) The m2r package August 3, 2017 13 / 20

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SLIDE 73

The parser

> m2_matrix(matrix(c(1,2,3,4), nrow = 2, ncol = 2)) [,1] [,2] [1,] 1 3 [2,] 2 4 M2 Matrix over ZZ[] > m2_matrix(matrix(c(1,2,3,4), nrow = 2, ncol = 2), code = TRUE) m2rintmatrix00000002 = matrix {{(1),(3)},{(2),(4)}} >

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SLIDE 74

The parser

> m2_matrix(matrix(c(1,2,3,4), nrow = 2, ncol = 2)) [,1] [,2] [1,] 1 3 [2,] 2 4 M2 Matrix over ZZ[] > m2_matrix(matrix(c(1,2,3,4), nrow = 2, ncol = 2), code = TRUE) m2rintmatrix00000002 = matrix {{(1),(3)},{(2),(4)}} > m2("m2rintmatrix00000002 = matrix {{(1),(3)},{(2),(4)}}") [1] "map((ZZ)ˆ2,(ZZ)ˆ2,{{1, 3}, {2, 4}})" >

Christopher O’Neill (UC Davis) The m2r package August 3, 2017 13 / 20

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SLIDE 75

The parser

> m2_matrix(matrix(c(1,2,3,4), nrow = 2, ncol = 2)) [,1] [,2] [1,] 1 3 [2,] 2 4 M2 Matrix over ZZ[] > m2_matrix(matrix(c(1,2,3,4), nrow = 2, ncol = 2), code = TRUE) m2rintmatrix00000002 = matrix {{(1),(3)},{(2),(4)}} > m2("m2rintmatrix00000002 = matrix {{(1),(3)},{(2),(4)}}") [1] "map((ZZ)ˆ2,(ZZ)ˆ2,{{1, 3}, {2, 4}})" > m2_parse("map((ZZ)ˆ2,(ZZ)ˆ2,{{1, 3}, {2, 4}})") [,1] [,2] [1,] 1 3 [2,] 2 4 M2 Matrix over ZZ[]

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SLIDE 76

The parser

Parsing "map((ZZ)ˆ2,(ZZ)ˆ2,{{1, 3}, {2, 4}})"

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SLIDE 77

The parser

Parsing "map((ZZ)ˆ2,(ZZ)ˆ2,{{1, 3}, {2, 4}})" > m2_tokenize("map((ZZ)ˆ2,(ZZ)ˆ2,{{1, 3}, {2, 4}})") [1] "map" "(" "(" "ZZ" ")" "ˆ" "2" "," "(" "ZZ" [11] ")" "ˆ" "2" "," "{" "{" "1" "," "3" "}" [21] "," "{" "2" "," "4" "}" "}" ")"

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SLIDE 78

The parser

Parsing "map((ZZ)ˆ2,(ZZ)ˆ2,{{1, 3}, {2, 4}})" > m2_tokenize("map((ZZ)ˆ2,(ZZ)ˆ2,{{1, 3}, {2, 4}})") [1] "map" "(" "(" "ZZ" ")" "ˆ" "2" "," "(" "ZZ" [11] ")" "ˆ" "2" "," "{" "{" "1" "," "3" "}" [21] "," "{" "2" "," "4" "}" "}" ")"

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SLIDE 79

The parser

Parsing "map((ZZ)ˆ2,(ZZ)ˆ2,{{1, 3}, {2, 4}})" > m2_tokenize("map((ZZ)ˆ2,(ZZ)ˆ2,{{1, 3}, {2, 4}})") [1] "map" "(" "(" "ZZ" ")" "ˆ" "2" "," "(" "ZZ" [11] ")" "ˆ" "2" "," "{" "{" "1" "," "3" "}" [21] "," "{" "2" "," "4" "}" "}" ")"

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SLIDE 80

The parser

Parsing "map((ZZ)ˆ2,(ZZ)ˆ2,{{1, 3}, {2, 4}})" > m2_tokenize("map((ZZ)ˆ2,(ZZ)ˆ2,{{1, 3}, {2, 4}})") [1] "map" "(" "(" "ZZ" ")" "ˆ" "2" "," "(" "ZZ" [11] ")" "ˆ" "2" "," "{" "{" "1" "," "3" "}" [21] "," "{" "2" "," "4" "}" "}" ")"

Christopher O’Neill (UC Davis) The m2r package August 3, 2017 14 / 20

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SLIDE 81

The parser

Parsing "map((ZZ)ˆ2,(ZZ)ˆ2,{{1, 3}, {2, 4}})" > m2_tokenize("map((ZZ)ˆ2,(ZZ)ˆ2,{{1, 3}, {2, 4}})") [1] "map" "(" "(" "ZZ" ")" "ˆ" "2" "," "(" "ZZ" [11] ")" "ˆ" "2" "," "{" "{" "1" "," "3" "}" [21] "," "{" "2" "," "4" "}" "}" ")"

Christopher O’Neill (UC Davis) The m2r package August 3, 2017 14 / 20

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SLIDE 82

The parser

Parsing "map((ZZ)ˆ2,(ZZ)ˆ2,{{1, 3}, {2, 4}})" > m2_tokenize("map((ZZ)ˆ2,(ZZ)ˆ2,{{1, 3}, {2, 4}})") [1] "map" "(" "(" "ZZ" ")" "ˆ" "2" "," "(" "ZZ" [11] ")" "ˆ" "2" "," "{" "{" "1" "," "3" "}" [21] "," "{" "2" "," "4" "}" "}" ")"

Christopher O’Neill (UC Davis) The m2r package August 3, 2017 14 / 20

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SLIDE 83

The parser

Parsing "map((ZZ)ˆ2,(ZZ)ˆ2,{{1, 3}, {2, 4}})" > m2_tokenize("map((ZZ)ˆ2,(ZZ)ˆ2,{{1, 3}, {2, 4}})") [1] "map" "(" "(" "ZZ" ")" "ˆ" "2" "," "(" "ZZ" [11] ")" "ˆ" "2" "," "{" "{" "1" "," "3" "}" [21] "," "{" "2" "," "4" "}" "}" ")"

Christopher O’Neill (UC Davis) The m2r package August 3, 2017 14 / 20

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SLIDE 84

The parser

Parsing "map((ZZ)ˆ2,(ZZ)ˆ2,{{1, 3}, {2, 4}})" > m2_tokenize("map((ZZ)ˆ2,(ZZ)ˆ2,{{1, 3}, {2, 4}})") [1] "map" "(" "(" "ZZ" ")" "ˆ" "2" "," "(" "ZZ" [11] ")" "ˆ" "2" "," "{" "{" "1" "," "3" "}" [21] "," "{" "2" "," "4" "}" "}" ")"

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SLIDE 85

The parser

>

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SLIDE 86

The parser

> m2_parse(m2("x")) M2 Symbol: x >

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SLIDE 87

The parser

> m2_parse(m2("x")) M2 Symbol: x > m2_parse(m2("ZZ")) M2 Ring: ZZ[], grevlex order >

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SLIDE 88

The parser

> m2_parse(m2("x")) M2 Symbol: x > m2_parse(m2("ZZ")) M2 Ring: ZZ[], grevlex order > m2("m2rintring00000002 = QQ[x,y,z,MonomialOrder=>{GRevLex=>3}]") [1] "QQ(monoid[x..z, Degrees => {3:1}, Heft => {1}, MonomialOrder => VerticalList{MonomialSize => 32, GRevLex => {3:1}, Position => Up}, DegreeRank => 1])" >

Christopher O’Neill (UC Davis) The m2r package August 3, 2017 15 / 20

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SLIDE 89

The parser

> m2_parse(m2("x")) M2 Symbol: x > m2_parse(m2("ZZ")) M2 Ring: ZZ[], grevlex order > m2("m2rintring00000002 = QQ[x,y,z,MonomialOrder=>{GRevLex=>3}]") [1] "QQ(monoid[x..z, Degrees => {3:1}, Heft => {1}, MonomialOrder => VerticalList{MonomialSize => 32, GRevLex => {3:1}, Position => Up}, DegreeRank => 1])" > m2_parse(m2("m2rintring00000002")) M2 Ring: QQ[x,y,z], grevlex order >

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SLIDE 90

The parser

> m2_parse(m2("x")) M2 Symbol: x > m2_parse(m2("ZZ")) M2 Ring: ZZ[], grevlex order > m2("m2rintring00000002 = QQ[x,y,z,MonomialOrder=>{GRevLex=>3}]") [1] "QQ(monoid[x..z, Degrees => {3:1}, Heft => {1}, MonomialOrder => VerticalList{MonomialSize => 32, GRevLex => {3:1}, Position => Up}, DegreeRank => 1])" > m2_parse(m2("m2rintring00000002")) M2 Ring: QQ[x,y,z], grevlex order > m2("ideal({xˆ2+2*x,2*x+3})") [1] "ideal map((m2rintring00000002)ˆ1,(m2rintring00000002)ˆ{{-2}, {-1}},{{xˆ2+2*x, 2*x+3}})"

Christopher O’Neill (UC Davis) The m2r package August 3, 2017 15 / 20

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SLIDE 91

Parser “extensibility”

> m2("ideal({xˆ2+2*x,2*x+3})") [1] "ideal map((m2rintring00000002)ˆ1,(m2rintring00000002)ˆ{{-2}, {-1}},{{xˆ2+2*x, 2*x+3}})"

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SLIDE 92

Parser “extensibility”

> m2("ideal({xˆ2+2*x,2*x+3})") [1] "ideal map((m2rintring00000002)ˆ1,(m2rintring00000002)ˆ{{-2}, {-1}},{{xˆ2+2*x, 2*x+3}})" m2_parse_function.m2_map <- function(x) { R1 <- x[[1]] R2 <- x[[2]] . . . m2_structure( mat, m2_name = "", m2_class = "m2_matrix", m2_meta = list( ring = R1 ), base_class = "matrix" ) }

Christopher O’Neill (UC Davis) The m2r package August 3, 2017 16 / 20

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SLIDE 93

Parser “extensibility”

> m2("ideal({xˆ2+2*x,2*x+3})") [1] "ideal map((m2rintring00000002)ˆ1,(m2rintring00000002)ˆ{{-2}, {-1}},{{xˆ2+2*x, 2*x+3}})"

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SLIDE 94

Parser “extensibility”

> m2("ideal({xˆ2+2*x,2*x+3})") [1] "ideal map((m2rintring00000002)ˆ1,(m2rintring00000002)ˆ{{-2}, {-1}},{{xˆ2+2*x, 2*x+3}})" m2_parse_function.m2_ideal <- function(x) { m2_structure( m2_name = "", m2_class = "m2_ideal", m2_meta = list( ring = m2_meta(x[[1]], "ring"), gens = structure(x[[1]][1,], class = "mpolyList") ) ) }

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SLIDE 95

m2r is now in the cloud!

>

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SLIDE 96

m2r is now in the cloud!

> library("m2r") Loading required package: mpoly Loading required package: stringr please cite mpoly if you use it; see citation("mpoly") M2 not found; defaulting to cloud. Use set_m2r_path("/path/to/m2") to run M2 locally. >

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SLIDE 97

m2r is now in the cloud!

> library("m2r") Loading required package: mpoly Loading required package: stringr please cite mpoly if you use it; see citation("mpoly") M2 not found; defaulting to cloud. Use set_m2r_path("/path/to/m2") to run M2 locally. > start_m2() Connecting to M2 in the cloud... done. >

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SLIDE 98

m2r is now in the cloud!

> library("m2r") Loading required package: mpoly Loading required package: stringr please cite mpoly if you use it; see citation("mpoly") M2 not found; defaulting to cloud. Use set_m2r_path("/path/to/m2") to run M2 locally. > start_m2() Connecting to M2 in the cloud... done. > m2("a = 5") [1] "5" >

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SLIDE 99

m2r is now in the cloud!

> library("m2r") Loading required package: mpoly Loading required package: stringr please cite mpoly if you use it; see citation("mpoly") M2 not found; defaulting to cloud. Use set_m2r_path("/path/to/m2") to run M2 locally. > start_m2() Connecting to M2 in the cloud... done. > m2("a = 5") [1] "5" > m2("a") [1] "5"

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SLIDE 100

m2r is now in the cloud!

> library("m2r") Loading required package: mpoly Loading required package: stringr please cite mpoly if you use it; see citation("mpoly") M2 not found; defaulting to cloud. Use set_m2r_path("/path/to/m2") to run M2 locally. > start_m2() Connecting to M2 in the cloud... done. > m2("a = 5") [1] "5" > m2("a") [1] "5"

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SLIDE 101

m2r is now in the cloud!

> library("m2r") Loading required package: mpoly Loading required package: stringr please cite mpoly if you use it; see citation("mpoly") M2 not found; defaulting to cloud. Use set_m2r_path("/path/to/m2") to run M2 locally. > start_m2() Connecting to M2 in the cloud... done. > m2("a = 5") [1] "5" > m2("a") [1] "5"

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SLIDE 102

m2r is now in the cloud!

> library("m2r") Loading required package: mpoly Loading required package: stringr please cite mpoly if you use it; see citation("mpoly") M2 not found; defaulting to cloud. Use set_m2r_path("/path/to/m2") to run M2 locally. > start_m2() Connecting to M2 in the cloud... done. > m2("a = 5") [1] "5" > m2("a") [1] "5"

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SLIDE 103

m2r is now in the cloud!

> library("m2r") Loading required package: mpoly Loading required package: stringr please cite mpoly if you use it; see citation("mpoly") M2 not found; defaulting to cloud. Use set_m2r_path("/path/to/m2") to run M2 locally. > start_m2() Connecting to M2 in the cloud... done. > m2("a = 5") [1] "5" > m2("a") [1] "5"

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SLIDE 104

More fancy features out there: reference functions

>

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SLIDE 105

More fancy features out there: reference functions

> R <- ring("x", "y", "z", coefring = "QQ") M2 Ring: QQ[x,y,z], grevlex order >

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SLIDE 106

More fancy features out there: reference functions

> R <- ring("x", "y", "z", coefring = "QQ") M2 Ring: QQ[x,y,z], grevlex order > (I <- ideal("xˆ2", "x*y", "xˆ3*yˆ2")) M2 Ideal of ring QQ[x,y,z] (grevlex) with generators : < xˆ2, x y, xˆ3 yˆ2 > >

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SLIDE 107

More fancy features out there: reference functions

> R <- ring("x", "y", "z", coefring = "QQ") M2 Ring: QQ[x,y,z], grevlex order > (I <- ideal("xˆ2", "x*y", "xˆ3*yˆ2")) M2 Ideal of ring QQ[x,y,z] (grevlex) with generators : < xˆ2, x y, xˆ3 yˆ2 > > gb(I) x y xˆ2 >

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SLIDE 108

More fancy features out there: reference functions

> R <- ring("x", "y", "z", coefring = "QQ") M2 Ring: QQ[x,y,z], grevlex order > (I <- ideal("xˆ2", "x*y", "xˆ3*yˆ2")) M2 Ideal of ring QQ[x,y,z] (grevlex) with generators : < xˆ2, x y, xˆ3 yˆ2 > > gb(I) x y xˆ2 > (J <- ideal.("xˆ2", "x*y", "xˆ3*yˆ2")) M2 Pointer Object ExternalString : ideal map((m2rintring00000001)ˆ1,(m2rin... M2 Name : m2rintideal00000004 M2 Class : Ideal (Type) >

Christopher O’Neill (UC Davis) The m2r package August 3, 2017 18 / 20

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SLIDE 109

More fancy features out there: reference functions

> R <- ring("x", "y", "z", coefring = "QQ") M2 Ring: QQ[x,y,z], grevlex order > (I <- ideal("xˆ2", "x*y", "xˆ3*yˆ2")) M2 Ideal of ring QQ[x,y,z] (grevlex) with generators : < xˆ2, x y, xˆ3 yˆ2 > > gb(I) x y xˆ2 > (J <- ideal.("xˆ2", "x*y", "xˆ3*yˆ2")) M2 Pointer Object ExternalString : ideal map((m2rintring00000001)ˆ1,(m2rin... M2 Name : m2rintideal00000004 M2 Class : Ideal (Type) > gb(J) x y xˆ2

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SLIDE 110

More fancy features out there: reference functions

> R <- ring("x", "y", "z", coefring = "QQ") M2 Ring: QQ[x,y,z], grevlex order > (I <- ideal("xˆ2", "x*y", "xˆ3*yˆ2")) M2 Ideal of ring QQ[x,y,z] (grevlex) with generators : < xˆ2, x y, xˆ3 yˆ2 > > gb(I) x y xˆ2 > (J <- ideal.("xˆ2", "x*y", "xˆ3*yˆ2")) M2 Pointer Object ExternalString : ideal map((m2rintring00000001)ˆ1,(m2rin... M2 Name : m2rintideal00000004 M2 Class : Ideal (Type) > gb(J) x y xˆ2 m2_parse(J) − → I

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SLIDE 111

Thank you MRC!

Christopher O’Neill (UC Davis) The m2r package August 3, 2017 19 / 20

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SLIDE 112

References

  • D. Kahle, C. O’Neill, and J. Sommars (2017)

A computer algebra system for R: Macaulay2 and the m2r package

  • submitted. Available at [arXiv:1706.07797].
  • D. Grayson and M. Stillman (2006)

Macaulay2, a software system for research in algebraic geometry, available at http://www.math.uiuc.edu/Macaulay2/.

  • D. Kahle (2013)

mpoly: Multivariate polynomials in R The R Journal 5 (1), 162–170. R Core Team (2014) R: A language and environment for statistical computing R Foundation for Statistical Computing, Vienna, Austria

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SLIDE 113

References

  • D. Kahle, C. O’Neill, and J. Sommars (2017)

A computer algebra system for R: Macaulay2 and the m2r package

  • submitted. Available at [arXiv:1706.07797].
  • D. Grayson and M. Stillman (2006)

Macaulay2, a software system for research in algebraic geometry, available at http://www.math.uiuc.edu/Macaulay2/.

  • D. Kahle (2013)

mpoly: Multivariate polynomials in R The R Journal 5 (1), 162–170. R Core Team (2014) R: A language and environment for statistical computing R Foundation for Statistical Computing, Vienna, Austria Thanks!

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SLIDE 114

YOU should request lots of features!

https://github.com/coneill-math/m2r

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SLIDE 115

YOU should request lots of features!

https://github.com/coneill-math/m2r

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SLIDE 116

YOU should request lots of features!

https://github.com/coneill-math/m2r

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SLIDE 117

YOU should request lots of features!

https://github.com/coneill-math/m2r

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