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# complexrank -- compute the complex rank

## Synopsis

• Usage:
complexrank F
• Inputs:
• F, , 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