Changeset 6b589f in git
 Timestamp:
 Apr 19, 2007, 5:10:38 PM (17 years ago)
 Branches:
 (u'fiekerDuVal', '117eb8c30fc9e991c4decca4832b1d19036c4c65')(u'spielwiese', 'b52fc4b2495505785981d640dcf7eb3e456778ef')
 Children:
 eff00fb66b5cef03a89608c1e8adcc5cf36ade3c
 Parents:
 bb9a5a2a1df0eacab065576595d4d0635e0ecae1
 File:

 1 edited
Legend:
 Unmodified
 Added
 Removed

Singular/LIB/alexpoly.lib
rbb9a5a r6b589f 1 version="$Id: alexpoly.lib,v 1.1 4 20060802 10:14:06Singular Exp $";1 version="$Id: alexpoly.lib,v 1.15 20070419 15:10:38 Singular Exp $"; 2 2 category="Singularities"; 3 3 info=" … … 17 17 alexanderpolynomial(f); Alexander polynomial of f 18 18 semigroup(f); calculates generators for the semigroup of f 19 proximitymatrix(f); calculates the proximity matrix of f 19 20 multseq2charexp(v); converts multiplicity sequence to characteristic exponents 20 21 charexp2multseq(v); converts characteristic exponents to multiplicity sequence … … 2222 2223 } 2223 2224 2225 proc proximitymatrix (def INPUT) 2226 "USAGE: proximitymatrix(INPUT); INPUT poly or list or intmat 2227 ASSUME: INPUT is either a REDUCED bivariate polynomial defining a plane curve singularity, 2228 or the output of @code{hnexpansion(f[,\"ess\"])}, or the list @code{hne} in 2229 the ring created by @code{hnexpansion(f[,\"ess\"])}, or the output of 2230 @code{develop(f)} resp. of @code{extdevelop(f,n)}, or a list containing 2231 the contact matrix and a list of integer vectors with the characteristic exponents 2232 of the branches of a plane curve singularity, or an integer vector containing the 2233 characteristic exponents of an irreducible plane curve singularity, or the resolution 2234 graph of a plane curve singularity (i.e. the output of resolutiongraph or 2235 the first entry in the output of totalmultiplicities). 2236 RETURN: list, of three integer matrices. The first one is the proximity matrix of 2237 the plane curve defined by the INPUT, i.e. the entry i,j is 1 if the 2238 infinitely near point corresponding to row i is proximate to the infinitely 2239 near point corresponding to row j. The second integer matrix is the incidence matrix of the 2240 resolution graph of the plane curve. The entry on the diagonal in row i is s1 2241 if s is the number of points proximate to the infinitely near point corresponding 2242 to the ith row in the matrix. The third integer matrix is the incidence matrix of 2243 the Enriques diagram of the plane curve singularity, i.e. each row corresponds to 2244 an infinitely near point in the minimal standard resolution, including the 2245 strict transforms of the branches, the diagonal element gives 2246 the level of the point, and the entry i,j is 1 if row i is proximate to row j. 2247 NOTE: In case the HamburgerNoether expansion of the curve f is needed 2248 for other purposes as well it is better to calculate this first 2249 with the aid of @code{hnexpansion} and use it as input instead of 2250 the polynomial itself. 2251 @* 2252 If you are not sure whether the INPUT polynomial is reduced or not, use 2253 @code{squarefree(INPUT)} as input instead. 2254 @* 2255 If the input is a smooth curve, then the output will consist of 2256 three onebyone zero matrices. 2257 @* 2258 For the definitions of the computed objects see e.g. the book 2259 Eduardo CasasAlvero, Singularities of Plane Curves. 2260 SEE ALSO: develop, hnexpansion, totalmultiplicities, alexanderpolynomial 2261 EXAMPLE: example proximitymatrix; shows an example 2262 " 2263 { 2264 ///////// Decide on the type of input. ////////// 2265 if (typeof(INPUT)=="intmat") 2266 { 2267 intmat resgr=INPUT; 2268 } 2269 else 2270 { 2271 intmat resgr=totalmultiplicities(INPUT)[1]; 2272 } 2273 //////// Deal with the case of a smooth curve //////////////// 2274 if (size(resgr)==1) 2275 { 2276 return(list(intmat(intvec(1),1,1),intmat(intvec(1),1,1),intmat(intvec(0),1,1))); 2277 } 2278 //////// Calculate the proximity resolution graph //////////// 2279 int i,j; 2280 int n=nrows(resgr); 2281 intvec diag; // Diagonal of the Enriques diagram. 2282 int si; // number of points proximate to the point corresponding to the ith row 2283 // Adjust the weights of the nodes corresponding to strict transforms so 2284 // as if there had been one additional blow up. 2285 for (i=1;i<=n;i++) 2286 { 2287 // Check if the row corresponds to an exceptional divisor ... 2288 if (resgr[i,i]<0) 2289 { 2290 j=1; 2291 while ((resgr[i,j]==0) or (i==j)) 2292 { 2293 j++; 2294 } 2295 resgr[i,i]=resgr[j,j]+1; 2296 } 2297 } 2298 // Set the weights in the resolution graph to the blowing up level in the resolution. 2299 for (i=1;i<=n;i++) 2300 { 2301 resgr[i,i]=resgr[i,i]1; 2302 diag[i]=resgr[i,i]; // The level of the corresponding infinitely near point. 2303 } 2304 // Replace the weights in the resolution graph by 2305 // s1, where s is the number of points which are proximate to the point. 2306 for (i=1;i<=n;i++) 2307 { 2308 si=1; 2309 // Find the points of higher weight which are connected to the ith row. 2310 for (j=i+1;j<=n;j++) 2311 { 2312 // If the point in row j is connected to the ith row, then all the points of 2313 // weight resgr[i,i]+1 to weight resgr[j,j] in the corresponding subgraph 2314 // are proximate to the point of the ith row. This number is just resgr[j,j]resgr[i,i]. 2315 if ((resgr[i,j]!=0) and (resgr[j,j]>0)) 2316 { 2317 si=si(resgr[j,j]resgr[i,i]); 2318 } 2319 } 2320 resgr[i,i]=si; 2321 } 2322 /////////////// Calculate the proximity matrix /////////////////// 2323 intmat proximity=proxgauss(resgr); 2324 intmat enriques=proximity; 2325 for (i=1;i<=nrows(enriques);i++) 2326 { 2327 enriques[i,i]=diag[i]; 2328 } 2329 return(list(proximity,resgr,enriques)); 2330 } 2331 example 2332 { 2333 "EXAMPLE:"; 2334 echo=2; 2335 ring r=0,(x,y),ls; 2336 poly f1=(y2x3)^24x5yx7; 2337 poly f2=y2x3; 2338 poly f3=y3x2; 2339 list proximity=proximitymatrix(f1*f2*f3); 2340 /// The proximity matrix P /// 2341 print(proximity[1]); 2342 /// The proximity resolution graph N /// 2343 print(proximity[2]); 2344 /// They satisfy N=transpose(P)*P /// 2345 print(transpose(proximity[1])*proximity[1]); 2346 /// The incidence matrix of the Enriques diagram /// 2347 print(proximity[3]); 2348 /// If M is the matrix of multiplicities and TM the matrix of total 2349 /// multiplicities of the singularity, then M=P*TM. 2350 /// We therefore calculate the (total) multiplicities. Note that 2351 /// they have to be slightly extended. 2352 list MULT=extend_multiplicities(totalmultiplicities(f1*f2*f3)); 2353 intmat TM=MULT[1]; // Total multiplicites. 2354 intmat M=MULT[2]; // Multiplicities. 2355 /// Check: MP*TM=0. 2356 Mproximity[1]*TM; 2357 /// Check: inverse(P)*MTM=0. 2358 intmat_inverse(proximity[1])*MTM; 2359 } 2360 2361 static proc addmultiplrows (intmat M, int i, int j, int ki, int kj) 2362 "USAGE: addmultiplrows(M,i,j,ki,kj); intmat M, int i,j,ki,kj 2363 RETURN: intmat, replaces the jth row in M by kitimes the ith row plus 2364 kj times the jth 2365 NOTE: This procedure is only for internal use; it is called in intmat_inverse 2366 and proxgauss. 2367 " 2368 { 2369 for (int k=1;k<=ncols(M);k++) 2370 { 2371 M[j,k]=kj*M[j,k]+ki*M[i,k]; 2372 } 2373 return(M); 2374 } 2375 2376 2377 static proc proxgauss (intmat M) 2378 "USAGE: proxgauss(M); intmat M 2379 ASSUME: M is the output of proximity_resgr 2380 RETURN: intmat, replaces the jth row in M by kitimes the ith row plus 2381 kj times the jth 2382 NOTE: This procedure is only for internal use; it is called in intmat_inverse. 2383 " 2384 { 2385 int i; 2386 int n=nrows(M); 2387 if (n==1) 2388 { 2389 M[1,1]=1; 2390 return(M); 2391 } 2392 else 2393 { 2394 if (M[n,n]<0) 2395 { 2396 M=addmultiplrows(M,n,n,1,0); 2397 } 2398 for (i=n1;i>=1;i) 2399 { 2400 if (M[i,n]!=0) 2401 { 2402 M=addmultiplrows(M,n,i,M[i,n],M[n,n]); 2403 } 2404 } 2405 intmat N[n1][n1]=M[1..n1,1..n1]; 2406 N=proxgauss(N); 2407 M[1..n1,1..n1]=N[1..n1,1..n1]; 2408 return(M); 2409 } 2410 } 2411 2412 proc extend_multiplicities (list rg) 2413 "USAGE: extend_multiplicities(rg); list rg 2414 ASSUME: rg is the output of the procedure totalmultiplicities 2415 RETURN: list, the first entry is an integer matrix containing the total 2416 multiplicities and the second entry is an integer matrix containing 2417 the multiplicies of the resolution given by rg, where the zeros 2418 corresponding to the strict transforms of the branches of the curve 2419 have been replaced by the (total) multiplicities of the infinitely near 2420 point corresponding to one further blow up for each branch. 2421 KEYWORDS: total multiplicities; multiplicity sequence; resolution graph 2422 EXAMPLE: example extend_multiplicities; shows an example 2423 " 2424 { 2425 intmat resgr,tm,mt=rg[1],rg[2],rg[3]; 2426 int i,j; 2427 for (i=1;i<=nrows(resgr);i++) 2428 { 2429 if (resgr[i,i]<0) 2430 { 2431 j=1; 2432 while ((resgr[i,j]==0) or (i==j)) 2433 { 2434 j++; 2435 } 2436 tm[i,1..ncols(tm)]=tm[j,1..ncols(tm)]; 2437 tm[i,resgr[i,i]]=tm[i,resgr[i,i]]+1; 2438 mt[i,resgr[i,i]]=1; 2439 } 2440 } 2441 return(list(tm,mt)); 2442 } 2443 example 2444 { 2445 "EXAMPLE:"; 2446 echo=2; 2447 ring r=0,(x,y),ls; 2448 poly f1=(y2x3)^24x5yx7; 2449 poly f2=y2x3; 2450 poly f3=y3x2; 2451 // Calculate the resolution graph and the (total) multiplicities of f1*f2*f3. 2452 list RESGR=totalmultiplicities(f1*f2*f3); 2453 // Extend the (total) multiplicities. 2454 list MULT=extend_multiplicities(RESGR); 2455 // Compare the total multiplicities. 2456 RESGR[2]; 2457 MULT[1]; 2458 // Compare the multiplicities. 2459 RESGR[3]; 2460 MULT[2]; 2461 } 2462 2463 proc intmat_inverse (intmat M) 2464 "USAGE: intmat_inverse(M); intmat M 2465 ASSUME: M is a lower triangular integer matrix with diagonal entries 1 or 1 2466 RETURN: intmat, the inverse of M 2467 KEYWORDS: integer matrix, inverse 2468 EXAMPLE: example intmat_inverse; shows an example 2469 " 2470 { 2471 int i,j,k; 2472 int n=nrows(M); 2473 intmat U[n][n]; 2474 for (i=1;i<=n;i++) 2475 { 2476 U[i,i]=1; 2477 } 2478 for (i=1;i<=n;i++) 2479 { 2480 if (M[i,i]==1) 2481 { 2482 M=addmultiplrows(M,i,i,1,0); 2483 U=addmultiplrows(U,i,i,1,0); 2484 } 2485 for (j=i+1;j<=n;j++) 2486 { 2487 U=addmultiplrows(U,i,j,M[j,i],M[i,i]); 2488 M=addmultiplrows(M,i,j,M[j,i],M[i,i]); 2489 } 2490 } 2491 return(U); 2492 } 2493 example 2494 { 2495 "EXAMPLE:";echo=2; 2496 intmat M[5][5]=1,0,0,0,0,1,1,0,0,0,2,1,1,0,0,3,1,1,1,0,4,1,1,1,1 ; 2497 intmat U=intmat_inverse(M); 2498 print(U); 2499 print(U*M); 2500 }
Note: See TracChangeset
for help on using the changeset viewer.