If X has codimension 1, then we intersect X with a randomly choosen 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 Sn with a fix point on {1,...,n }, i.e.
which approachs 1-exp(-1) = 0.63.... 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 0.053612 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 0.180678 seconds o10 = ideal (x + 48x , x + 26x , x - 30x , x - 19x , x + 30x ) 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 0.568803 seconds o15 = 58 |