A.1.4 Long coefficients

The following innocent example produces in its standard basis extremely long coefficients in char 0 for the lexicographical ordering. But a very small deformation does not (the undeformed example is degenerated with respect to the Newton boundary). This example demonstrates that it might be wise, for complicated examples, to do the calculation first in positive char (e.g., 32003). It has been shown, that in complicated examples, more than 95 percent of the time needed for a standard basis computation is used in the computation of the coefficients (in char 0). The representation of long integers with real is demonstrated.

timer = 1; // activate the timer ring R0 = 0,(x,y),lp; poly f = x5+y11+xy9+x3y9; ideal i = jacob(f); ideal i1 = i,i[1]*i[2]; // undeformed ideal ideal i2 = i,i[1]*i[2]+1/1000000*x5y8; // deformation of i1 i1; i2; ==> i1[1]=5x4+3x2y9+y9 ==> i1[2]=9x3y8+9xy8+11y10 ==> i1[3]=45x7y8+27x5y17+45x5y8+55x4y10+36x3y17+33x2y19+9xy17+11y19 ==> i2[1]=5x4+3x2y9+y9 ==> i2[2]=9x3y8+9xy8+11y10 ==> i2[3]=45x7y8+27x5y17+45000001/1000000x5y8+55x4y10+36x3y17+33x2y19+9xy17+1\ 1y19 ideal j = std(i1); j; ==> j[1]=264627y39+26244y35-1323135y30-131220y26+1715175y21+164025y17+1830125\ y16 ==> j[2]=12103947791971846719838321886393392913750065060875xy8-28639152114168\ 3198701331939250003266767738632875y38-31954402206909026926764622877573565\ 78554430672591y37+57436621420822663849721381265738895282846320y36+1657764\ 214948799497573918210031067353932439400y35+213018481589308191195677223898\ 98682697001205500y34+1822194158663066565585991976961565719648069806148y33\ -4701709279892816135156972313196394005220175y32-1351872269688192267600786\ 97600850686824231975y31-3873063305929810816961516976025038053001141375y30\ +1325886675843874047990382005421144061861290080000y29+1597720195476063141\ 9467945895542406089526966887310y28-26270181336309092660633348002625330426\ 7126525y27-7586082690893335269027136248944859544727953125y26-867853074106\ 49464602285843351672148965395945625y25-5545808143273594102173252331151835\ 700278863924745y24+19075563013460437364679153779038394895638325y23+548562\ 322715501761058348996776922561074021125y22+157465452677648386073957464715\ 68100780933983125y21-1414279129721176222978654235817359505555191156250y20\ -20711190069445893615213399650035715378169943423125y19+272942733337472665\ 573418092977905322984009750y18+789065115845334505801847294677413365720955\ 3750y17+63554897038491686787729656061044724651089803125y16-22099251729923\ 906699732244761028266074350255961625y14+147937139679655904353579489722585\ 91339027857296625y10 ==> j[3]=5x4+3x2y9+y9 // Compute average coefficient length (=51) by // - converting j[2] to a string in order to compute the number // of characters // - divide this by the number of monomials: size(string(j[2])) div size(j[2]); ==> 51 vdim(j); ==> 63 // For a better representation normalize the long coefficients // of the polynomial j[2] and map it to real: poly p=(1/12103947791971846719838321886393392913750065060875)*j[2]; ring R1=real,(x,y),lp; short=0; // force the long output format poly p=imap(R0,p); p; ==> x*y^8-(2.366e-02)*y^38-(2.640e-01)*y^37+(4.745e-06)*y^36+(1.370e-04)*y^35\ +(1.760e-03)*y^34+(1.505e-01)*y^33-(3.884e-07)*y^32-(1.117e-05)*y^31-(3.2\ 00e-04)*y^30+(1.095e-01)*y^29+(1.320e+00)*y^28-(2.170e-05)*y^27-(6.267e-0\ 4)*y^26-(7.170e-03)*y^25-(4.582e-01)*y^24+(1.576e-06)*y^23+(4.532e-05)*y^\ 22+(1.301e-03)*y^21-(1.168e-01)*y^20-(1.711e+00)*y^19+(2.255e-05)*y^18+(6\ .519e-04)*y^17+(5.251e-03)*y^16-(1.826e+00)*y^14+(1.222e+00)*y^10 // Compute a standard basis for the deformed ideal: setring R0; // return to the original ring R0 j = std(i2); j; ==> j[1]=y16 ==> j[2]=65610xy8+17393508y27+7223337y23+545292y19+6442040y18-119790y14+80190\ y10 ==> j[3]=5x4+3x2y9+y9 vdim(j); ==> 40