realrank -- compute the real rank

Synopsis

• Usage:

  realrank F

• Inputs:
  • F, a ring element, a binary form $F\in K[x,y]$ of degree $d$, where $K$ is either $\mathbb{Q}$ or a polynomial ring over $\mathbb{Q}$ (currently, with only one variable)
• Optional inputs:
  • Limit => ..., default value infinity, set a bound for the rank
  • QepcadOptions => ..., default value (20000000,10000), set the number of cells in the garbage collected space
  • Range => ..., default value (-infinity,infinity), can be assigned an interval
  • Verbose => ..., default value false, request verbose feedback
• Outputs:
  • an integer, the real 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 $l_1,\ldots,l_r$ are real linear forms and $c_1,\ldots,c_r\in\mathbb{R}$

Description

This method requires generally the program QEPCAD to be installed. Source code and installation instructions for it are available at Downloading and Installing QEPCAD.

Below we compute the real rank of a binary form of degree 7.

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

i2 : F = 2*x^7+7*x^6*y+168*x^5*y^2+140*x^4*y^3+70*x^3*y^4+21*x^2*y^5+56*x*y^6+4*y^7

       7     6        5 2       4 3      3 4      2 5        6     7
o2 = 2x  + 7x y + 168x y  + 140x y  + 70x y  + 21x y  + 56x*y  + 4y

o2 : QQ[x..y]

i3 : realrank F

o3 = 5

In the case when the coefficient ring $K$ contains a variable, say $u$, then the method returns a value $r$ if the real rank of $F$ is $r$ for all the real values of $u$ in the range specified by the option realrank(...,Range=>...). An error is thrown if the answer is not uniform.

i4 : Ru := QQ[u][x,y];

i5 : F = u*x^4*y+2*x^2*y^3

        4      2 3
o5 = u*x y + 2x y

o5 : QQ[u][x..y]

i6 : realrank(F,Range=>(0,infinity))

o6 = 3

i7 : realrank(F,Range=>[-1,0])

o7 = 3

i8 : realrank(F,Range=>(-infinity,-1))

o8 = 3

See also

• complexrank -- compute the complex rank