If X has codimension 1, then we intersect X with a randomly chosen line, and hope that the decomposition of the the intersection contains a K-rational point. If n=degree X then the probability P that this happens, is the proportion of permutations in $S_n$ with a fix point on $\{1,\ldots,n \}$, i.e. $$P=\sum_{j=1}^n (-1)^{j-1} binomial(n,j)(n-j)!/n! = 1-1/2+1/3! + \ldots $$ which approaches $1-exp(-1) = 0.63\ldots$. Thus a probabilistic approach works.
For higher codimension we first project X birationally onto a hypersurface Y, and find a point on Y. Then we take the preimage of this point.
i1 : p=nextPrime(random(2*10^4)) o1 = 107 |
i2 : kk=ZZ/p;R=kk[x_0..x_3]; |
i4 : I=minors(4,random(R^5,R^{4:-1}));
o4 : Ideal of R
|
i5 : codim I, degree I o5 = (2, 10) o5 : Sequence |
i6 : time randomKRationalPoint(I)
-- used 0.453173 seconds
o6 = ideal (x - 53x , x + 8x , x - 4x )
2 3 1 3 0 3
o6 : Ideal of R
|
i7 : R=kk[x_0..x_5]; |
i8 : I=minors(3,random(R^5,R^{3:-1}));
o8 : Ideal of R
|
i9 : codim I, degree I o9 = (3, 10) o9 : Sequence |
i10 : time randomKRationalPoint(I)
-- used 0.744571 seconds
o10 = ideal (x - 27x , x - 16x , x - 9x , x + 44x , x - 52x )
4 5 3 5 2 5 1 5 0 5
o10 : Ideal of R
|
The claim that $63 \%$ of the intersections contain a K-rational point can be experimentally tested:
i11 : p=10007,kk=ZZ/p,R=kk[x_0..x_2] o11 = (10007, kk, R) o11 : Sequence |
i12 : n=5; sum(1..n,j->(-1)^(j-1)*binomial(n,j)*(n-j)!/n!)+0.0 o13 = .633333333333333 o13 : RR (of precision 53) |
i14 : I=ideal random(n,R); o14 : Ideal of R |
i15 : time (#select(apply(100,i->(degs=apply(decompose(I+ideal random(1,R)),c->degree c);
#select(degs,d->d==1))),f->f>0))
-- used 3.67775 seconds
o15 = 58
|
The object randomKRationalPoint is a method function.