complexrank -- compute the complex rank

Synopsis

Usage:

complexrank F
Inputs:
- F, a ring element, a binary form $F\in K[x,y]$ of degree $d$, where $K$ is either $\mathbb{Q}$, or more generally a field
Optional inputs:
- Limit => ..., default value infinity, set a bound for the rank
Outputs:
- an integer, the rank of $F$, i.e., the minimum integer $r$ such that there is a decomposition $F = c_1\,(l_1)^d+\cdots+c_r\,(l_r)^d$ where $l_1,\ldots,l_r\in\bar{K}[x,y]$ are linear forms and $c_1,\ldots,c_r\in\bar{K}$

Description

This method provides a quick way to calculate the complex rank of a binary form as an application of the methods apolar(RingElement,ZZ) and discriminant(RingElement).

i1 : R := QQ[x,y];

i2 : F = 325699392019820093805938500473136959995883*x^11-5810907570924644857232186920803498012892938*x^10*y+65819917752061707843768328400359649501719860*x^9*y^2-519457154316395169830396776661486079064173600*x^8*y^3+1705429425321816258526777767700378341505324800*x^7*y^4-3810190868583760635545828188931628645390528000*x^6*y^5+9250941324308079844692884039573393626015320480*x^5*y^6-9323164714263069666482962682446368124512793200*x^4*y^7+1072684515031339121680779290598231336889158000*x^3*y^8-66208958025372412656331871291180685863962950*x^2*y^9-3357470237827984950448384820635661305324565*x*y^10+2036327846200712576945384935680953020530520*y^11

                                                11  
o2 = 325699392019820093805938500473136959995883x   -
     ------------------------------------------------------------------------
                                                 10   
     5810907570924644857232186920803498012892938x  y +
     ------------------------------------------------------------------------
                                                  9 2  
     65819917752061707843768328400359649501719860x y  -
     ------------------------------------------------------------------------
                                                   8 3  
     519457154316395169830396776661486079064173600x y  +
     ------------------------------------------------------------------------
                                                    7 4  
     1705429425321816258526777767700378341505324800x y  -
     ------------------------------------------------------------------------
                                                    6 5  
     3810190868583760635545828188931628645390528000x y  +
     ------------------------------------------------------------------------
                                                    5 6  
     9250941324308079844692884039573393626015320480x y  -
     ------------------------------------------------------------------------
                                                    4 7  
     9323164714263069666482962682446368124512793200x y  +
     ------------------------------------------------------------------------
                                                    3 8  
     1072684515031339121680779290598231336889158000x y  -
     ------------------------------------------------------------------------
                                                  2 9  
     66208958025372412656331871291180685863962950x y  -
     ------------------------------------------------------------------------
                                                   10  
     3357470237827984950448384820635661305324565x*y   +
     ------------------------------------------------------------------------
                                                 11
     2036327846200712576945384935680953020530520y

o2 : QQ[x..y]

i3 : complexrank F

o3 = 8

Ways to use complexrank :

complexrank(RingElement)

For the programmer

The object complexrank is a method function with options.

complexrank -- compute the complex rank

Synopsis

Description

See also

Ways to use complexrank :

For the programmer