Download Local cohomology and its applications by Gennady Lybeznik PDF

By Gennady Lybeznik

Includes displays from the overseas workshop on neighborhood cohomology held in Guanajuato, Mexico.

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Example text

1. Evaluate a given polynomial in 700 random points. 2. 14. Count singular quadrics. -C. Graf v. 18. Count quadrics with dim > 0 singular locus function findk(n,p,k,c) //Search until k singular examples of codim at most c are found, //p prime number, n dimension K := FiniteField(p); R := PolynomialRing(K,n); trials := 0; found := 0; while found lt k do Q := Ideal([Random(2,R,0)]); if c ge n - Dimension(Q+JacobianIdeal(Basis(Q))) then found := found + 1; else trials := trials + 1; end if; end while; print "Trails:",trials; return trials; end function; k := 50; time L1 := [[p,findk(4,p,k,2)] : p in [5,7,11]]; L1; time findk(4,5,50,2); time findk(4,7,50,2); time findk(4,11,50,2); function slope(L) //calculate slope of regression line by //formula form [2] p.

3 using p-adic Newton iteration. -C. Graf v. Bothmer / Finite Field Experiments 35 -- (in our application this division will not have a remainder) divn = (M,n) -> ( matrix apply(rank target M, i-> apply(rank source M,j-> M_j_i//n))) -- invert number mod n invn = (i,n) -> ( c := gcdCoefficients(i,n); if c#0 == 1 then c#1 else "error" ) -- invert a matrix mod n -- M a square matrix over ZZ -- (if M is not invertible mod n, then 0 is returned) invMatn = (M,n) -> ( Mn := modn(M,n); MQQ := sub(Mn,QQ); detM = sub(det Mn,QQ); modn(invn(sub(detM,ZZ),n)*sub(detM*MQQ^-1,ZZ),n) ) With this we can implement Newton iteration.

3. R := PolynomialRing(IntegerRing(),2); //two variables //equations F := 176*x^2+148*x*y+301*y^2-742*x+896*y+768; G := -25*x*y+430*y^2+33*x+1373*y+645; I := Ideal([F,G]); for p in PrimesUpTo(41) do print p, allPoints(I,p); end for; // x coordinate chineseList([[2,1],[13,5],[31,7],[37,14],[41,0]]); // y coordinate chineseList([[2,0],[13,10],[31,22],[37,18],[41,23]]); //test the solution Evaluate(F,[138949,-526048]); Evaluate(G,[138949,-526048]); /*take (a,n) and calculate a solution to r = as mod n such that r,s < sqrt(n).

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