Bernstein-Sato polynomials of the simple and unimodal singularities

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Polynomials

singularity

$n$

$b_f(s)$

$A_1$

s^2 + 3/2*s + 1/2

comment: $f=x^{2}$, with $\mu=1$ and $\operatorname{lct}(f)=\frac{1}{2}$; $b_f(s)=\left(s+\frac{1}{2}\right)(s+1)$.

$A_1$

s^2 + 2*s + 1

comment: $f=x^{2}+y^2$, with $\mu=1$ and $\operatorname{lct}(f)=1$; $b_f(s)=(s+1)^{2}$.

$A_1$

s^2 + 5/2*s + 3/2

comment: $f=x^{2}+y^2+z^2$, with $\mu=1$ and $\operatorname{lct}(f)=1$; $b_f(s)=(s+1)\left(s+\frac{3}{2}\right)$.

$A_2$

s^3 + 2*s^2 + 11/9*s + 2/9

comment: $f=x^{3}$, with $\mu=2$ and $\operatorname{lct}(f)=\frac{1}{3}$; $b_f(s)=\left(s+\frac{1}{3}\right)\left(s+\frac{2}{3}\right)(s+1)$.

$A_2$

s^3 + 3*s^2 + 107/36*s + 35/36

comment: $f=x^{3}+y^2$, with $\mu=2$ and $\operatorname{lct}(f)=\frac{5}{6}$; $b_f(s)=\left(s+\frac{5}{6}\right)(s+1)\left(s+\frac{7}{6}\right)$.

$A_2$

s^3 + 4*s^2 + 47/9*s + 20/9

comment: $f=x^{3}+y^2+z^2$, with $\mu=2$ and $\operatorname{lct}(f)=1$; $b_f(s)=(s+1)\left(s+\frac{4}{3}\right)\left(s+\frac{5}{3}\right)$.

$A_3$

s^4 + 5/2*s^3 + 35/16*s^2 + 25/32*s + 3/32

comment: $f=x^{4}$, with $\mu=3$ and $\operatorname{lct}(f)=\frac{1}{4}$; $b_f(s)=\left(s+\frac{1}{4}\right)\left(s+\frac{1}{2}\right)\left(s+\frac{3}{4}\right)(s+1)$.

$A_3$

s^4 + 4*s^3 + 95/16*s^2 + 31/8*s + 15/16

comment: $f=x^{4}+y^2$, with $\mu=3$ and $\operatorname{lct}(f)=\frac{3}{4}$; $b_f(s)=\left(s+\frac{3}{4}\right)(s+1)^{2}\left(s+\frac{5}{4}\right)$.

$A_3$

s^4 + 11/2*s^3 + 179/16*s^2 + 319/32*s + 105/32

comment: $f=x^{4}+y^2+z^2$, with $\mu=3$ and $\operatorname{lct}(f)=1$; $b_f(s)=(s+1)\left(s+\frac{5}{4}\right)\left(s+\frac{3}{2}\right)\left(s+\frac{7}{4}\right)$.

$A_4$

s^5 + 3*s^4 + 17/5*s^3 + 9/5*s^2 + 274/625*s + 24/625

comment: $f=x^{5}$, with $\mu=4$ and $\operatorname{lct}(f)=\frac{1}{5}$; $b_f(s)=\left(s+\frac{1}{5}\right)\left(s+\frac{2}{5}\right)\left(s+\frac{3}{5}\right)\left(s+\frac{4}{5}\right)(s+1)$.

$A_4$

s^5 + 5*s^4 + 99/10*s^3 + 97/10*s^2 + 47009/10000*s + 9009/10000

comment: $f=x^{5}+y^2$, with $\mu=4$ and $\operatorname{lct}(f)=\frac{7}{10}$; $b_f(s)=\left(s+\frac{7}{10}\right)\left(s+\frac{9}{10}\right)(s+1)\left(s+\frac{11}{10}\right)\left(s+\frac{13}{10}\right)$.

$A_4$

s^5 + 7*s^4 + 97/5*s^3 + 133/5*s^2 + 11274/625*s + 3024/625

comment: $f=x^{5}+y^2+z^2$, with $\mu=4$ and $\operatorname{lct}(f)=1$; $b_f(s)=(s+1)\left(s+\frac{6}{5}\right)\left(s+\frac{7}{5}\right)\left(s+\frac{8}{5}\right)\left(s+\frac{9}{5}\right)$.

$A_5$

s^6 + 7/2*s^5 + 175/36*s^4 + 245/72*s^3 + 203/162*s^2 + 49/216*s + 5/324

comment: $f=x^{6}$, with $\mu=5$ and $\operatorname{lct}(f)=\frac{1}{6}$; $b_f(s)=\left(s+\frac{1}{6}\right)\left(s+\frac{1}{3}\right)\left(s+\frac{1}{2}\right)\left(s+\frac{2}{3}\right)\left(s+\frac{5}{6}\right)(s+1)$.

$A_5$

s^6 + 6*s^5 + 535/36*s^4 + 175/9*s^3 + 4591/324*s^2 + 883/162*s + 70/81

comment: $f=x^{6}+y^2$, with $\mu=5$ and $\operatorname{lct}(f)=\frac{2}{3}$; $b_f(s)=\left(s+\frac{2}{3}\right)\left(s+\frac{5}{6}\right)(s+1)^{2}\left(s+\frac{7}{6}\right)\left(s+\frac{4}{3}\right)$.

$A_5$

s^6 + 17/2*s^5 + 1075/36*s^4 + 3995/72*s^3 + 18631/324*s^2 + 20417/648*s + 385/54

comment: $f=x^{6}+y^2+z^2$, with $\mu=5$ and $\operatorname{lct}(f)=1$; $b_f(s)=(s+1)\left(s+\frac{7}{6}\right)\left(s+\frac{4}{3}\right)\left(s+\frac{3}{2}\right)\left(s+\frac{5}{3}\right)\left(s+\frac{11}{6}\right)$.

$A_6$

s^7 + 4*s^6 + 46/7*s^5 + 40/7*s^4 + 967/343*s^3 + 268/343*s^2 + 13068/117649*s + 720/117649

comment: $f=x^{7}$, with $\mu=6$ and $\operatorname{lct}(f)=\frac{1}{7}$; $b_f(s)=\left(s+\frac{1}{7}\right)\left(s+\frac{2}{7}\right)\left(s+\frac{3}{7}\right)\left(s+\frac{4}{7}\right)\left(s+\frac{5}{7}\right)\left(s+\frac{6}{7}\right)(s+1)$.

$A_6$

s^7 + 7*s^6 + 583/28*s^5 + 955/28*s^4 + 182317/5488*s^3 + 105559/5488*s^2 + 46136019/7529536*s + 6235515/7529536

comment: $f=x^{7}+y^2$, with $\mu=6$ and $\operatorname{lct}(f)=\frac{9}{14}$; $b_f(s)=\left(s+\frac{9}{14}\right)\left(s+\frac{11}{14}\right)\left(s+\frac{13}{14}\right)(s+1)\left(s+\frac{15}{14}\right)\left(s+\frac{17}{14}\right)\left(s+\frac{19}{14}\right)$.

$A_6$

s^7 + 10*s^6 + 298/7*s^5 + 100*s^4 + 48007/343*s^3 + 40030/343*s^2 + 6314664/117649*s + 1235520/117649

comment: $f=x^{7}+y^2+z^2$, with $\mu=6$ and $\operatorname{lct}(f)=1$; $b_f(s)=(s+1)\left(s+\frac{8}{7}\right)\left(s+\frac{9}{7}\right)\left(s+\frac{10}{7}\right)\left(s+\frac{11}{7}\right)\left(s+\frac{12}{7}\right)\left(s+\frac{13}{7}\right)$.

$A_7$

s^8 + 9/2*s^7 + 273/32*s^6 + 567/64*s^5 + 22449/4096*s^4 + 16821/8192*s^3 + 29531/65536*s^2 + 6849/131072*s + 315/131072

comment: $f=x^{8}$, with $\mu=7$ and $\operatorname{lct}(f)=\frac{1}{8}$; $b_f(s)=\left(s+\frac{1}{8}\right)\left(s+\frac{1}{4}\right)\left(s+\frac{3}{8}\right)\left(s+\frac{1}{2}\right)\left(s+\frac{5}{8}\right)\left(s+\frac{3}{4}\right)\left(s+\frac{7}{8}\right)(s+1)$.

$A_7$

s^8 + 8*s^7 + 889/32*s^6 + 875/16*s^5 + 273329/4096*s^4 + 52913/1024*s^3 + 1624663/65536*s^2 + 220695/32768*s + 51975/65536

comment: $f=x^{8}+y^2$, with $\mu=7$ and $\operatorname{lct}(f)=\frac{5}{8}$; $b_f(s)=\left(s+\frac{5}{8}\right)\left(s+\frac{3}{4}\right)\left(s+\frac{7}{8}\right)(s+1)^{2}\left(s+\frac{9}{8}\right)\left(s+\frac{5}{4}\right)\left(s+\frac{11}{8}\right)$.

$A_7$

s^8 + 23/2*s^7 + 1841/32*s^6 + 10465/64*s^5 + 1182769/4096*s^4 + 2657627/8192*s^3 + 14838507/65536*s^2 + 11762775/131072*s + 2027025/131072

comment: $f=x^{8}+y^2+z^2$, with $\mu=7$ and $\operatorname{lct}(f)=1$; $b_f(s)=(s+1)\left(s+\frac{9}{8}\right)\left(s+\frac{5}{4}\right)\left(s+\frac{11}{8}\right)\left(s+\frac{3}{2}\right)\left(s+\frac{13}{8}\right)\left(s+\frac{7}{4}\right)\left(s+\frac{15}{8}\right)$.

$A_8$

s^9 + 5*s^8 + 290/27*s^7 + 350/27*s^6 + 21091/2187*s^5 + 3325/729*s^4 + 723680/531441*s^3 + 130300/531441*s^2 + 114064/4782969*s + 4480/4782969

comment: $f=x^{9}$, with $\mu=8$ and $\operatorname{lct}(f)=\frac{1}{9}$; $b_f(s)=\left(s+\frac{1}{9}\right)\left(s+\frac{2}{9}\right)\left(s+\frac{1}{3}\right)\left(s+\frac{4}{9}\right)\left(s+\frac{5}{9}\right)\left(s+\frac{2}{3}\right)\left(s+\frac{7}{9}\right)\left(s+\frac{8}{9}\right)(s+1)$.

$A_8$

s^9 + 9*s^8 + 965/27*s^7 + 2219/27*s^6 + 2109569/17496*s^5 + 2047381/17496*s^4 + 638695055/8503056*s^3 + 87134951/2834352*s^2 + 8911558585/1224440064*s + 929553625/1224440064

comment: $f=x^{9}+y^2$, with $\mu=8$ and $\operatorname{lct}(f)=\frac{11}{18}$; $b_f(s)=\left(s+\frac{11}{18}\right)\left(s+\frac{13}{18}\right)\left(s+\frac{5}{6}\right)\left(s+\frac{17}{18}\right)(s+1)\left(s+\frac{19}{18}\right)\left(s+\frac{7}{6}\right)\left(s+\frac{23}{18}\right)\left(s+\frac{25}{18}\right)$.

$A_8$

s^9 + 13*s^8 + 2018/27*s^7 + 6734/27*s^6 + 1164163/2187*s^5 + 1647919/2187*s^4 + 375923456/531441*s^3 + 225666428/531441*s^2 + 707390800/4782969*s + 108908800/4782969

comment: $f=x^{9}+y^2+z^2$, with $\mu=8$ and $\operatorname{lct}(f)=1$; $b_f(s)=(s+1)\left(s+\frac{10}{9}\right)\left(s+\frac{11}{9}\right)\left(s+\frac{4}{3}\right)\left(s+\frac{13}{9}\right)\left(s+\frac{14}{9}\right)\left(s+\frac{5}{3}\right)\left(s+\frac{16}{9}\right)\left(s+\frac{17}{9}\right)$.

$A_9$

s^10 + 11/2*s^9 + 66/5*s^8 + 363/20*s^7 + 157773/10000*s^6 + 180411/20000*s^5 + 341693/100000*s^4 + 16819/20000*s^3 + 1594197/12500000*s^2 + 66429/6250000*s + 567/1562500

comment: $f=x^{10}$, with $\mu=9$ and $\operatorname{lct}(f)=\frac{1}{10}$; $b_f(s)=\left(s+\frac{1}{10}\right)\left(s+\frac{1}{5}\right)\left(s+\frac{3}{10}\right)\left(s+\frac{2}{5}\right)\left(s+\frac{1}{2}\right)\left(s+\frac{3}{5}\right)\left(s+\frac{7}{10}\right)\left(s+\frac{4}{5}\right)\left(s+\frac{9}{10}\right)(s+1)$.

$A_9$

s^10 + 10*s^9 + 447/10*s^8 + 588/5*s^7 + 2016273/10000*s^6 + 1176819/5000*s^5 + 4735217/25000*s^4 + 324196/3125*s^3 + 231278661/6250000*s^2 + 24251661/3125000*s + 567567/781250

comment: $f=x^{10}+y^2$, with $\mu=9$ and $\operatorname{lct}(f)=\frac{3}{5}$; $b_f(s)=\left(s+\frac{3}{5}\right)\left(s+\frac{7}{10}\right)\left(s+\frac{4}{5}\right)\left(s+\frac{9}{10}\right)(s+1)^{2}\left(s+\frac{11}{10}\right)\left(s+\frac{6}{5}\right)\left(s+\frac{13}{10}\right)\left(s+\frac{7}{5}\right)$.

$A_9$

s^10 + 29/2*s^9 + 471/5*s^8 + 7221/20*s^7 + 9040773/10000*s^6 + 30906141/20000*s^5 + 182584943/100000*s^4 + 147244049/100000*s^3 + 9694951947/12500000*s^2 + 301184343/1250000*s + 26189163/781250

comment: $f=x^{10}+y^2+z^2$, with $\mu=9$ and $\operatorname{lct}(f)=1$; $b_f(s)=(s+1)\left(s+\frac{11}{10}\right)\left(s+\frac{6}{5}\right)\left(s+\frac{13}{10}\right)\left(s+\frac{7}{5}\right)\left(s+\frac{3}{2}\right)\left(s+\frac{8}{5}\right)\left(s+\frac{17}{10}\right)\left(s+\frac{9}{5}\right)\left(s+\frac{19}{10}\right)$.

$A_{10}$

s^11 + 6*s^10 + 175/11*s^9 + 270/11*s^8 + 32493/1331*s^7 + 21798/1331*s^6 + 1212685/161051*s^5 + 380130/161051*s^4 + 9568916/19487171*s^3 + 1247256/19487171*s^2 + 120543840/25937424601*s + 3628800/25937424601

comment: $f=x^{11}$, with $\mu=10$ and $\operatorname{lct}(f)=\frac{1}{11}$; $b_f(s)=\left(s+\frac{1}{11}\right)\left(s+\frac{2}{11}\right)\left(s+\frac{3}{11}\right)\left(s+\frac{4}{11}\right)\left(s+\frac{5}{11}\right)\left(s+\frac{6}{11}\right)\left(s+\frac{7}{11}\right)\left(s+\frac{8}{11}\right)\left(s+\frac{9}{11}\right)\left(s+\frac{10}{11}\right)(s+1)$.

$A_{10}$

s^11 + 11*s^10 + 2405/44*s^9 + 7125/44*s^8 + 3383559/10648*s^7 + 4617249/10648*s^6 + 2163653645/5153632*s^5 + 1486046425/5153632*s^4 + 686746253791/4988715776*s^3 + 217004133501/4988715776*s^2 + 217434345380955/26559922791424*s + 18460681477875/26559922791424

comment: $f=x^{11}+y^2$, with $\mu=10$ and $\operatorname{lct}(f)=\frac{13}{22}$; $b_f(s)=\left(s+\frac{13}{22}\right)\left(s+\frac{15}{22}\right)\left(s+\frac{17}{22}\right)\left(s+\frac{19}{22}\right)\left(s+\frac{21}{22}\right)(s+1)\left(s+\frac{23}{22}\right)\left(s+\frac{25}{22}\right)\left(s+\frac{27}{22}\right)\left(s+\frac{29}{22}\right)\left(s+\frac{31}{22}\right)$.

$A_{10}$

s^11 + 16*s^10 + 1275/11*s^9 + 5520/11*s^8 + 1920093/1331*s^7 + 3848208/1331*s^6 + 663854665/161051*s^5 + 673248880/161051*s^4 + 57589621116/19487171*s^3 + 27027135936/19487171*s^2 + 10086271796160/25937424601*s + 1279935820800/25937424601

comment: $f=x^{11}+y^2+z^2$, with $\mu=10$ and $\operatorname{lct}(f)=1$; $b_f(s)=(s+1)\left(s+\frac{12}{11}\right)\left(s+\frac{13}{11}\right)\left(s+\frac{14}{11}\right)\left(s+\frac{15}{11}\right)\left(s+\frac{16}{11}\right)\left(s+\frac{17}{11}\right)\left(s+\frac{18}{11}\right)\left(s+\frac{19}{11}\right)\left(s+\frac{20}{11}\right)\left(s+\frac{21}{11}\right)$.

$A_{11}$

s^12 + 13/2*s^11 + 2717/144*s^10 + 9295/288*s^9 + 249821/6912*s^8 + 42757/1536*s^7 + 44990231/2985984*s^6 + 34345025/5971968*s^5 + 164301709/107495424*s^4 + 58917287/214990848*s^3 + 1676701/53747712*s^2 + 430105/214990848*s + 1925/35831808

comment: $f=x^{12}$, with $\mu=11$ and $\operatorname{lct}(f)=\frac{1}{12}$; $b_f(s)=\left(s+\frac{1}{12}\right)\left(s+\frac{1}{6}\right)\left(s+\frac{1}{4}\right)\left(s+\frac{1}{3}\right)\left(s+\frac{5}{12}\right)\left(s+\frac{1}{2}\right)\left(s+\frac{7}{12}\right)\left(s+\frac{2}{3}\right)\left(s+\frac{3}{4}\right)\left(s+\frac{5}{6}\right)\left(s+\frac{11}{12}\right)(s+1)$.

$A_{11}$

s^12 + 12*s^11 + 9449/144*s^10 + 15565/72*s^9 + 3302981/6912*s^8 + 645029/864*s^7 + 2523665507/2985984*s^6 + 347616995/497664*s^5 + 44955309289/107495424*s^4 + 4753404337/26873856*s^3 + 5391514169/107495424*s^2 + 460083925/53747712*s + 2977975/4478976

comment: $f=x^{12}+y^2$, with $\mu=11$ and $\operatorname{lct}(f)=\frac{7}{12}$; $b_f(s)=\left(s+\frac{7}{12}\right)\left(s+\frac{2}{3}\right)\left(s+\frac{3}{4}\right)\left(s+\frac{5}{6}\right)\left(s+\frac{11}{12}\right)(s+1)^{2}\left(s+\frac{13}{12}\right)\left(s+\frac{7}{6}\right)\left(s+\frac{5}{4}\right)\left(s+\frac{4}{3}\right)\left(s+\frac{17}{12}\right)$.

$A_{11}$

s^12 + 35/2*s^11 + 20141/144*s^10 + 194425/288*s^9 + 15147341/6912*s^8 + 69676915/13824*s^7 + 25147102343/2985984*s^6 + 61510571815/5971968*s^5 + 983660827369/107495424*s^4 + 1238183559305/214990848*s^3 + 262000691741/107495424*s^2 + 133879793635/214990848*s + 1301375075/17915904

comment: $f=x^{12}+y^2+z^2$, with $\mu=11$ and $\operatorname{lct}(f)=1$; $b_f(s)=(s+1)\left(s+\frac{13}{12}\right)\left(s+\frac{7}{6}\right)\left(s+\frac{5}{4}\right)\left(s+\frac{4}{3}\right)\left(s+\frac{17}{12}\right)\left(s+\frac{3}{2}\right)\left(s+\frac{19}{12}\right)\left(s+\frac{5}{3}\right)\left(s+\frac{7}{4}\right)\left(s+\frac{11}{6}\right)\left(s+\frac{23}{12}\right)$.

$A_{12}$

s^13 + 7*s^12 + 287/13*s^11 + 539/13*s^10 + 113421/2197*s^9 + 98637/2197*s^8 + 10387421/371293*s^7 + 4680137/371293*s^6 + 256624522/62748517*s^5 + 58921324/62748517*s^4 + 1562596392/10604499373*s^3 + 157377024/10604499373*s^2 + 19802759040/23298085122481*s + 479001600/23298085122481

comment: $f=x^{13}$, with $\mu=12$ and $\operatorname{lct}(f)=\frac{1}{13}$; $b_f(s)=\left(s+\frac{1}{13}\right)\left(s+\frac{2}{13}\right)\left(s+\frac{3}{13}\right)\left(s+\frac{4}{13}\right)\left(s+\frac{5}{13}\right)\left(s+\frac{6}{13}\right)\left(s+\frac{7}{13}\right)\left(s+\frac{8}{13}\right)\left(s+\frac{9}{13}\right)\left(s+\frac{10}{13}\right)\left(s+\frac{11}{13}\right)\left(s+\frac{12}{13}\right)(s+1)$.

$A_{12}$

s^13 + 13*s^12 + 2017/26*s^11 + 7315/26*s^10 + 24317931/35152*s^9 + 42806643/35152*s^8 + 9378237143/5940688*s^7 + 9064265705/5940688*s^6 + 17660019688267/16063620352*s^5 + 9367827977431/16063620352*s^4 + 1201749482686497/5429503678976*s^3 + 309005025194475/5429503678976*s^2 + 847865067446563425/95428956661682176*s + 60685940227460625/95428956661682176

comment: $f=x^{13}+y^2$, with $\mu=12$ and $\operatorname{lct}(f)=\frac{15}{26}$; $b_f(s)=\left(s+\frac{15}{26}\right)\left(s+\frac{17}{26}\right)\left(s+\frac{19}{26}\right)\left(s+\frac{21}{26}\right)\left(s+\frac{23}{26}\right)\left(s+\frac{25}{26}\right)(s+1)\left(s+\frac{27}{26}\right)\left(s+\frac{29}{26}\right)\left(s+\frac{31}{26}\right)\left(s+\frac{33}{26}\right)\left(s+\frac{35}{26}\right)\left(s+\frac{37}{26}\right)$.

$A_{12}$

s^13 + 19*s^12 + 2159/13*s^11 + 11495/13*s^10 + 7028901/2197*s^9 + 1403853/169*s^8 + 5912013437/371293*s^7 + 8470461725/371293*s^6 + 1532993573122/62748517*s^5 + 1211863838548/62748517*s^4 + 116164798868904/10604499373*s^3 + 44766453970080/10604499373*s^2 + 23080457713017600/23298085122481*s + 2490952020480000/23298085122481

comment: $f=x^{13}+y^2+z^2$, with $\mu=12$ and $\operatorname{lct}(f)=1$; $b_f(s)=(s+1)\left(s+\frac{14}{13}\right)\left(s+\frac{15}{13}\right)\left(s+\frac{16}{13}\right)\left(s+\frac{17}{13}\right)\left(s+\frac{18}{13}\right)\left(s+\frac{19}{13}\right)\left(s+\frac{20}{13}\right)\left(s+\frac{21}{13}\right)\left(s+\frac{22}{13}\right)\left(s+\frac{23}{13}\right)\left(s+\frac{24}{13}\right)\left(s+\frac{25}{13}\right)$.

$D_4$

s^4 + 4*s^3 + 53/9*s^2 + 34/9*s + 8/9

comment: $f=x^2y+y^{3}$, with $\mu=4$ and $\operatorname{lct}(f)=\frac{2}{3}$; $b_f(s)=\left(s+\frac{2}{3}\right)(s+1)^{2}\left(s+\frac{4}{3}\right)$.

$D_4$

s^4 + 11/2*s^3 + 401/36*s^2 + 709/72*s + 77/24

comment: $f=x^2y+y^{3}+z^2$, with $\mu=4$ and $\operatorname{lct}(f)=1$; $b_f(s)=(s+1)\left(s+\frac{7}{6}\right)\left(s+\frac{3}{2}\right)\left(s+\frac{11}{6}\right)$.

$D_5$

s^6 + 6*s^5 + 475/32*s^4 + 155/8*s^3 + 57609/4096*s^2 + 11017/2048*s + 3465/4096

comment: $f=x^2y+y^{4}$, with $\mu=5$ and $\operatorname{lct}(f)=\frac{5}{8}$; $b_f(s)=\left(s+\frac{5}{8}\right)\left(s+\frac{7}{8}\right)(s+1)^{2}\left(s+\frac{9}{8}\right)\left(s+\frac{11}{8}\right)$.

$D_5$

s^6 + 17/2*s^5 + 955/32*s^4 + 3545/64*s^3 + 234729/4096*s^2 + 256653/8192*s + 57915/8192

comment: $f=x^2y+y^{4}+z^2$, with $\mu=5$ and $\operatorname{lct}(f)=1$; $b_f(s)=(s+1)\left(s+\frac{9}{8}\right)\left(s+\frac{11}{8}\right)\left(s+\frac{3}{2}\right)\left(s+\frac{13}{8}\right)\left(s+\frac{15}{8}\right)$.

$D_6$

s^6 + 6*s^5 + 74/5*s^4 + 96/5*s^3 + 8629/625*s^2 + 3258/625*s + 504/625

comment: $f=x^2y+y^{5}$, with $\mu=6$ and $\operatorname{lct}(f)=\frac{3}{5}$; $b_f(s)=\left(s+\frac{3}{5}\right)\left(s+\frac{4}{5}\right)(s+1)^{2}\left(s+\frac{6}{5}\right)\left(s+\frac{7}{5}\right)$.

$D_6$

s^6 + 17/2*s^5 + 149/5*s^4 + 1103/20*s^3 + 568189/10000*s^2 + 123589/4000*s + 138567/20000

comment: $f=x^2y+y^{5}+z^2$, with $\mu=6$ and $\operatorname{lct}(f)=1$; $b_f(s)=(s+1)\left(s+\frac{11}{10}\right)\left(s+\frac{13}{10}\right)\left(s+\frac{3}{2}\right)\left(s+\frac{17}{10}\right)\left(s+\frac{19}{10}\right)$.

$D_7$

s^8 + 8*s^7 + 3997/144*s^6 + 1309/24*s^5 + 1376179/20736*s^4 + 265363/5184*s^3 + 8104967/331776*s^2 + 1093447/165888*s + 85085/110592

comment: $f=x^2y+y^{6}$, with $\mu=7$ and $\operatorname{lct}(f)=\frac{7}{12}$; $b_f(s)=\left(s+\frac{7}{12}\right)\left(s+\frac{3}{4}\right)\left(s+\frac{11}{12}\right)(s+1)^{2}\left(s+\frac{13}{12}\right)\left(s+\frac{5}{4}\right)\left(s+\frac{17}{12}\right)$.

$D_7$

s^8 + 23/2*s^7 + 8281/144*s^6 + 47033/288*s^5 + 5972659/20736*s^4 + 13397657/41472*s^3 + 74645651/331776*s^2 + 59021983/663552*s + 3380195/221184

comment: $f=x^2y+y^{6}+z^2$, with $\mu=7$ and $\operatorname{lct}(f)=1$; $b_f(s)=(s+1)\left(s+\frac{13}{12}\right)\left(s+\frac{5}{4}\right)\left(s+\frac{17}{12}\right)\left(s+\frac{3}{2}\right)\left(s+\frac{19}{12}\right)\left(s+\frac{7}{4}\right)\left(s+\frac{23}{12}\right)$.

$D_8$

s^8 + 8*s^7 + 194/7*s^6 + 380/7*s^5 + 3221/49*s^4 + 2468/49*s^3 + 2804332/117649*s^2 + 749040/117649*s + 86400/117649

comment: $f=x^2y+y^{7}$, with $\mu=8$ and $\operatorname{lct}(f)=\frac{4}{7}$; $b_f(s)=\left(s+\frac{4}{7}\right)\left(s+\frac{5}{7}\right)\left(s+\frac{6}{7}\right)(s+1)^{2}\left(s+\frac{8}{7}\right)\left(s+\frac{9}{7}\right)\left(s+\frac{10}{7}\right)$.

$D_8$

s^8 + 23/2*s^7 + 1609/28*s^6 + 9125/56*s^5 + 224821/784*s^4 + 502853/1568*s^3 + 1675754037/7529536*s^2 + 1319542065/15059072*s + 225655875/15059072

comment: $f=x^2y+y^{7}+z^2$, with $\mu=8$ and $\operatorname{lct}(f)=1$; $b_f(s)=(s+1)\left(s+\frac{15}{14}\right)\left(s+\frac{17}{14}\right)\left(s+\frac{19}{14}\right)\left(s+\frac{3}{2}\right)\left(s+\frac{23}{14}\right)\left(s+\frac{25}{14}\right)\left(s+\frac{27}{14}\right)$.

$D_9$

s^10 + 10*s^9 + 2859/64*s^8 + 939/8*s^7 + 6581211/32768*s^6 + 3830673/16384*s^5 + 786357731/4194304*s^4 + 107189987/1048576*s^3 + 155734209297/4294967296*s^2 + 16219194129/2147483648*s + 3011753745/4294967296

comment: $f=x^2y+y^{8}$, with $\mu=9$ and $\operatorname{lct}(f)=\frac{9}{16}$; $b_f(s)=\left(s+\frac{9}{16}\right)\left(s+\frac{11}{16}\right)\left(s+\frac{13}{16}\right)\left(s+\frac{15}{16}\right)(s+1)^{2}\left(s+\frac{17}{16}\right)\left(s+\frac{19}{16}\right)\left(s+\frac{21}{16}\right)\left(s+\frac{23}{16}\right)$.

$D_9$

s^10 + 29/2*s^9 + 6027/64*s^8 + 46173/128*s^7 + 29571675/32768*s^6 + 100969995/65536*s^5 + 7623686243/4194304*s^4 + 12273632257/8388608*s^3 + 3302983332369/4294967296*s^2 + 2047133682261/8589934592*s + 284010484275/8589934592

comment: $f=x^2y+y^{8}+z^2$, with $\mu=9$ and $\operatorname{lct}(f)=1$; $b_f(s)=(s+1)\left(s+\frac{17}{16}\right)\left(s+\frac{19}{16}\right)\left(s+\frac{21}{16}\right)\left(s+\frac{23}{16}\right)\left(s+\frac{3}{2}\right)\left(s+\frac{25}{16}\right)\left(s+\frac{27}{16}\right)\left(s+\frac{29}{16}\right)\left(s+\frac{31}{16}\right)$.

$D_{10}$

s^10 + 10*s^9 + 1205/27*s^8 + 3160/27*s^7 + 436681/2187*s^6 + 168770/729*s^5 + 98155385/531441*s^4 + 53189420/531441*s^3 + 168573484/4782969*s^2 + 34822640/4782969*s + 3203200/4782969

comment: $f=x^2y+y^{9}$, with $\mu=10$ and $\operatorname{lct}(f)=\frac{5}{9}$; $b_f(s)=\left(s+\frac{5}{9}\right)\left(s+\frac{2}{3}\right)\left(s+\frac{7}{9}\right)\left(s+\frac{8}{9}\right)(s+1)^{2}\left(s+\frac{10}{9}\right)\left(s+\frac{11}{9}\right)\left(s+\frac{4}{3}\right)\left(s+\frac{13}{9}\right)$.

$D_{10}$

s^10 + 29/2*s^9 + 5083/54*s^8 + 19453/54*s^7 + 15746885/17496*s^6 + 53670421/34992*s^5 + 3838059977/2125764*s^4 + 24651223231/17006112*s^3 + 930085129201/1224440064*s^2 + 574501803365/2448880128*s + 26469144625/816293376

comment: $f=x^2y+y^{9}+z^2$, with $\mu=10$ and $\operatorname{lct}(f)=1$; $b_f(s)=(s+1)\left(s+\frac{19}{18}\right)\left(s+\frac{7}{6}\right)\left(s+\frac{23}{18}\right)\left(s+\frac{25}{18}\right)\left(s+\frac{3}{2}\right)\left(s+\frac{29}{18}\right)\left(s+\frac{31}{18}\right)\left(s+\frac{11}{6}\right)\left(s+\frac{35}{18}\right)$.

$D_{11}$

s^12 + 12*s^11 + 5247/80*s^10 + 1727/8*s^9 + 38119389/80000*s^8 + 7429389/10000*s^7 + 5369014079/6400000*s^6 + 2211539517/3200000*s^5 + 10551677797221/25600000000*s^4 + 1110518157221/6400000000*s^3 + 4008623747499/81920000000*s^2 + 1699402563351/204800000000*s + 52404515163/81920000000

comment: $f=x^2y+y^{10}$, with $\mu=11$ and $\operatorname{lct}(f)=\frac{11}{20}$; $b_f(s)=\left(s+\frac{11}{20}\right)\left(s+\frac{13}{20}\right)\left(s+\frac{3}{4}\right)\left(s+\frac{17}{20}\right)\left(s+\frac{19}{20}\right)(s+1)^{2}\left(s+\frac{21}{20}\right)\left(s+\frac{23}{20}\right)\left(s+\frac{5}{4}\right)\left(s+\frac{27}{20}\right)\left(s+\frac{29}{20}\right)$.

$D_{11}$

s^12 + 35/2*s^11 + 11187/80*s^10 + 107943/160*s^9 + 175085889/80000*s^8 + 804673947/160000*s^7 + 53720829899/6400000*s^6 + 131226489163/12800000*s^5 + 232803307837221/25600000000*s^4 + 292508729119431/51200000000*s^3 + 988279184164059/409600000000*s^2 + 503833524948999/819200000000*s + 11723847276141/163840000000

comment: $f=x^2y+y^{10}+z^2$, with $\mu=11$ and $\operatorname{lct}(f)=1$; $b_f(s)=(s+1)\left(s+\frac{21}{20}\right)\left(s+\frac{23}{20}\right)\left(s+\frac{5}{4}\right)\left(s+\frac{27}{20}\right)\left(s+\frac{29}{20}\right)\left(s+\frac{3}{2}\right)\left(s+\frac{31}{20}\right)\left(s+\frac{33}{20}\right)\left(s+\frac{7}{4}\right)\left(s+\frac{37}{20}\right)\left(s+\frac{39}{20}\right)$.

$D_{12}$

s^12 + 12*s^11 + 721/11*s^10 + 2370/11*s^9 + 631713/1331*s^8 + 982296/1331*s^7 + 133752463/161051*s^6 + 109730730/161051*s^5 + 7880063996/19487171*s^4 + 3298817112/19487171*s^3 + 1230412716576/25937424601*s^2 + 207188755200/25937424601*s + 15850598400/25937424601

comment: $f=x^2y+y^{11}$, with $\mu=12$ and $\operatorname{lct}(f)=\frac{6}{11}$; $b_f(s)=\left(s+\frac{6}{11}\right)\left(s+\frac{7}{11}\right)\left(s+\frac{8}{11}\right)\left(s+\frac{9}{11}\right)\left(s+\frac{10}{11}\right)(s+1)^{2}\left(s+\frac{12}{11}\right)\left(s+\frac{13}{11}\right)\left(s+\frac{14}{11}\right)\left(s+\frac{15}{11}\right)\left(s+\frac{16}{11}\right)$.

$D_{12}$

s^12 + 35/2*s^11 + 6151/44*s^10 + 59315/88*s^9 + 2907723/1331*s^8 + 106779405/21296*s^7 + 43063775999/5153632*s^6 + 105005134675/10307264*s^5 + 44987261846291/4988715776*s^4 + 5126524236455/907039232*s^3 + 63231911908089561/26559922791424*s^2 + 32143735117106685/53119845582848*s + 3728200351341375/53119845582848

comment: $f=x^2y+y^{11}+z^2$, with $\mu=12$ and $\operatorname{lct}(f)=1$; $b_f(s)=(s+1)\left(s+\frac{23}{22}\right)\left(s+\frac{25}{22}\right)\left(s+\frac{27}{22}\right)\left(s+\frac{29}{22}\right)\left(s+\frac{31}{22}\right)\left(s+\frac{3}{2}\right)\left(s+\frac{35}{22}\right)\left(s+\frac{37}{22}\right)\left(s+\frac{39}{22}\right)\left(s+\frac{41}{22}\right)\left(s+\frac{43}{22}\right)$.

$E_6$

s^7 + 7*s^6 + 499/24*s^5 + 815/24*s^4 + 227563/6912*s^3 + 43627/2304*s^2 + 4461779/746496*s + 595595/746496

comment: $f=x^3+y^4$, with $\mu=6$ and $\operatorname{lct}(f)=\frac{7}{12}$; $b_f(s)=\left(s+\frac{7}{12}\right)\left(s+\frac{5}{6}\right)\left(s+\frac{11}{12}\right)(s+1)\left(s+\frac{13}{12}\right)\left(s+\frac{7}{6}\right)\left(s+\frac{17}{12}\right)$.

$E_6$

s^7 + 10*s^6 + 1021/24*s^5 + 2395/24*s^4 + 963403/6912*s^3 + 200275/1728*s^2 + 9913199/186624*s + 482885/46656

comment: $f=x^3+y^4+z^2$, with $\mu=6$ and $\operatorname{lct}(f)=1$; $b_f(s)=(s+1)\left(s+\frac{13}{12}\right)\left(s+\frac{4}{3}\right)\left(s+\frac{17}{12}\right)\left(s+\frac{19}{12}\right)\left(s+\frac{5}{3}\right)\left(s+\frac{23}{12}\right)$.

$E_7$

s^8 + 8*s^7 + 749/27*s^6 + 490/9*s^5 + 144613/2187*s^4 + 111244/2187*s^3 + 12854393/531441*s^2 + 3451930/531441*s + 400400/531441

comment: $f=x^3+xy^3$, with $\mu=7$ and $\operatorname{lct}(f)=\frac{5}{9}$; $b_f(s)=\left(s+\frac{5}{9}\right)\left(s+\frac{7}{9}\right)\left(s+\frac{8}{9}\right)(s+1)^{2}\left(s+\frac{10}{9}\right)\left(s+\frac{11}{9}\right)\left(s+\frac{13}{9}\right)$.

$E_7$

s^8 + 23/2*s^7 + 6209/108*s^6 + 35245/216*s^5 + 10061863/34992*s^4 + 22544879/69984*s^3 + 7619912873/34012224*s^2 + 6014612245/68024448*s + 343755125/22674816

comment: $f=x^3+xy^3+z^2$, with $\mu=7$ and $\operatorname{lct}(f)=1$; $b_f(s)=(s+1)\left(s+\frac{19}{18}\right)\left(s+\frac{23}{18}\right)\left(s+\frac{25}{18}\right)\left(s+\frac{3}{2}\right)\left(s+\frac{29}{18}\right)\left(s+\frac{31}{18}\right)\left(s+\frac{35}{18}\right)$.

$E_8$

s^9 + 9*s^8 + 1606/45*s^7 + 3682/45*s^6 + 2016371/16875*s^5 + 388871/3375*s^4 + 167056264/2278125*s^3 + 22542364/759375*s^2 + 17763519136/2562890625*s + 1820955136/2562890625

comment: $f=x^3+y^5$, with $\mu=8$ and $\operatorname{lct}(f)=\frac{8}{15}$; $b_f(s)=\left(s+\frac{8}{15}\right)\left(s+\frac{11}{15}\right)\left(s+\frac{13}{15}\right)\left(s+\frac{14}{15}\right)(s+1)\left(s+\frac{16}{15}\right)\left(s+\frac{17}{15}\right)\left(s+\frac{19}{15}\right)\left(s+\frac{22}{15}\right)$.

$E_8$

s^9 + 13*s^8 + 3361/45*s^7 + 2240/9*s^6 + 71563093/135000*s^5 + 101017651/135000*s^4 + 25514666269/36450000*s^3 + 7625652233/18225000*s^2 + 95137013446441/656100000000*s + 14562554020441/656100000000

comment: $f=x^3+y^5+z^2$, with $\mu=8$ and $\operatorname{lct}(f)=1$; $b_f(s)=(s+1)\left(s+\frac{31}{30}\right)\left(s+\frac{37}{30}\right)\left(s+\frac{41}{30}\right)\left(s+\frac{43}{30}\right)\left(s+\frac{47}{30}\right)\left(s+\frac{49}{30}\right)\left(s+\frac{53}{30}\right)\left(s+\frac{59}{30}\right)$.

$P_8$

s^5 + 7*s^4 + 173/9*s^3 + 233/9*s^2 + 154/9*s + 40/9

comment: $f=x^3+y^3+z^3$, with $\mu=8$ and $\operatorname{lct}(f)=1$; $b_f(s)=(s+1)^{2}\left(s+\frac{4}{3}\right)\left(s+\frac{5}{3}\right)\left(s+2\right)$.

$X_9$

s^6 + 6*s^5 + 235/16*s^4 + 75/4*s^3 + 841/64*s^2 + 153/32*s + 45/64

comment: $f=x^4+y^4$, with $\mu=9$ and $\operatorname{lct}(f)=\frac{1}{2}$; $b_f(s)=\left(s+\frac{1}{2}\right)\left(s+\frac{3}{4}\right)(s+1)^{2}\left(s+\frac{5}{4}\right)\left(s+\frac{3}{2}\right)$.

$X_9$

s^6 + 17/2*s^5 + 475/16*s^4 + 1745/32*s^3 + 889/16*s^2 + 953/32*s + 105/16

comment: $f=x^4+y^4+z^2$, with $\mu=9$ and $\operatorname{lct}(f)=1$; $b_f(s)=(s+1)^{2}\left(s+\frac{5}{4}\right)\left(s+\frac{3}{2}\right)\left(s+\frac{7}{4}\right)\left(s+2\right)$.

$J_{10}$

s^8 + 8*s^7 + 497/18*s^6 + 161/3*s^5 + 83209/1296*s^4 + 15673/324*s^3 + 29021/1296*s^2 + 3769/648*s + 35/54

comment: $f=x^3+y^6$, with $\mu=10$ and $\operatorname{lct}(f)=\frac{1}{2}$; $b_f(s)=\left(s+\frac{1}{2}\right)\left(s+\frac{2}{3}\right)\left(s+\frac{5}{6}\right)(s+1)^{2}\left(s+\frac{7}{6}\right)\left(s+\frac{4}{3}\right)\left(s+\frac{3}{2}\right)$.

$J_{10}$

s^8 + 23/2*s^7 + 2065/36*s^6 + 11669/72*s^5 + 183827/648*s^4 + 204113/648*s^3 + 140395/648*s^2 + 27347/324*s + 385/27

comment: $f=x^3+y^6+z^2$, with $\mu=10$ and $\operatorname{lct}(f)=1$; $b_f(s)=(s+1)^{2}\left(s+\frac{7}{6}\right)\left(s+\frac{4}{3}\right)\left(s+\frac{3}{2}\right)\left(s+\frac{5}{3}\right)\left(s+\frac{11}{6}\right)\left(s+2\right)$.

$E_{12}$

s^13 + 13*s^12 + 1627/21*s^11 + 5885/21*s^10 + 6355613/9261*s^9 + 1236191/1029*s^8 + 132616622941/85766121*s^7 + 18148662565/12252303*s^6 + 1901607233306/1801088541*s^5 + 995779480828/1801088541*s^4 + 164149089186952/794280046581*s^3 + 13819934079520/264760015527*s^2 + 58704593512614400/7355827511386641*s + 4101169346560000/7355827511386641

comment: $f=x^3+y^7$, with $\mu=12$ and $\operatorname{lct}(f)=\frac{10}{21}$; $b_f(s)=\left(s+\frac{10}{21}\right)\left(s+\frac{13}{21}\right)\left(s+\frac{16}{21}\right)\left(s+\frac{17}{21}\right)\left(s+\frac{19}{21}\right)\left(s+\frac{20}{21}\right)(s+1)\left(s+\frac{22}{21}\right)\left(s+\frac{23}{21}\right)\left(s+\frac{25}{21}\right)\left(s+\frac{26}{21}\right)\left(s+\frac{29}{21}\right)\left(s+\frac{32}{21}\right)$.

$E_{12}$

s^13 + 19*s^12 + 6971/42*s^11 + 18535/21*s^10 + 472330463/148176*s^9 + 1223361227/148176*s^8 + 21649656807229/1372257936*s^7 + 7725656228455/343064484*s^6 + 11070456123244331/461078666496*s^5 + 1243855214258987/65868380928*s^4 + 4330680213935164931/406671383849472*s^3 + 829045549761920845/203335691924736*s^2 + 28620144310388229606925/30129469486639681536*s + 3064202803551734255125/30129469486639681536

comment: $f=x^3+y^7+z^2$, with $\mu=12$ and $\operatorname{lct}(f)=\frac{41}{42}$; $b_f(s)=\left(s+\frac{41}{42}\right)(s+1)\left(s+\frac{47}{42}\right)\left(s+\frac{53}{42}\right)\left(s+\frac{55}{42}\right)\left(s+\frac{59}{42}\right)\left(s+\frac{61}{42}\right)\left(s+\frac{65}{42}\right)\left(s+\frac{67}{42}\right)\left(s+\frac{71}{42}\right)\left(s+\frac{73}{42}\right)\left(s+\frac{79}{42}\right)\left(s+\frac{85}{42}\right)$.

$E_{13}$

s^14 + 14*s^13 + 4069/45*s^12 + 5356/15*s^11 + 16250234/16875*s^10 + 6331468/3375*s^9 + 6200939602/2278125*s^8 + 6806015216/2278125*s^7 + 6386797252693/2562890625*s^6 + 1342685301386/854296875*s^5 + 9451182064961/12814453125*s^4 + 1066315933748/4271484375*s^3 + 92246623205884/1601806640625*s^2 + 1441920336752/177978515625*s + 93282565376/177978515625

comment: $f=x^3+xy^5$, with $\mu=13$ and $\operatorname{lct}(f)=\frac{7}{15}$; $b_f(s)=\left(s+\frac{7}{15}\right)\left(s+\frac{3}{5}\right)\left(s+\frac{11}{15}\right)\left(s+\frac{4}{5}\right)\left(s+\frac{13}{15}\right)\left(s+\frac{14}{15}\right)(s+1)^{2}\left(s+\frac{16}{15}\right)\left(s+\frac{17}{15}\right)\left(s+\frac{6}{5}\right)\left(s+\frac{19}{15}\right)\left(s+\frac{7}{5}\right)\left(s+\frac{23}{15}\right)$.

$E_{13}$

s^14 + 41/2*s^13 + 8749/45*s^12 + 40703/36*s^11 + 1216083869/270000*s^10 + 7020643201/540000*s^9 + 2044325267699/72900000*s^8 + 3347098704521/72900000*s^7 + 37619013859924783/656100000000*s^6 + 71314676816265311/1312200000000*s^5 + 63127470473545933/1640250000000*s^4 + 259085814918698027/13122000000000*s^3 + 45510502135399084489/6561000000000000*s^2 + 3920013727702097089/2624400000000000*s + 650564313686776489/4374000000000000

comment: $f=x^3+xy^5+z^2$, with $\mu=13$ and $\operatorname{lct}(f)=\frac{29}{30}$; $b_f(s)=\left(s+\frac{29}{30}\right)(s+1)\left(s+\frac{11}{10}\right)\left(s+\frac{37}{30}\right)\left(s+\frac{13}{10}\right)\left(s+\frac{41}{30}\right)\left(s+\frac{43}{30}\right)\left(s+\frac{3}{2}\right)\left(s+\frac{47}{30}\right)\left(s+\frac{49}{30}\right)\left(s+\frac{17}{10}\right)\left(s+\frac{53}{30}\right)\left(s+\frac{19}{10}\right)\left(s+\frac{61}{30}\right)$.

$E_{14}$

s^15 + 15*s^14 + 15029/144*s^13 + 64337/144*s^12 + 72760961/55296*s^11 + 156142987/55296*s^10 + 217888909927/47775744*s^9 + 29961293479/5308416*s^8 + 593878963644785/110075314176*s^7 + 438389716290071/110075314176*s^6 + 5947280787525259/2641807540224*s^5 + 2527326944400167/2641807540224*s^4 + 675081748063307953/2282521714753536*s^3 + 47764165306548529/760840571584512*s^2 + 671946143728171235/82170781731127296*s + 40473942576942875/82170781731127296

comment: $f=x^3+y^8$, with $\mu=14$ and $\operatorname{lct}(f)=\frac{11}{24}$; $b_f(s)=\left(s+\frac{11}{24}\right)\left(s+\frac{7}{12}\right)\left(s+\frac{17}{24}\right)\left(s+\frac{19}{24}\right)\left(s+\frac{5}{6}\right)\left(s+\frac{11}{12}\right)\left(s+\frac{23}{24}\right)(s+1)\left(s+\frac{25}{24}\right)\left(s+\frac{13}{12}\right)\left(s+\frac{7}{6}\right)\left(s+\frac{29}{24}\right)\left(s+\frac{31}{24}\right)\left(s+\frac{17}{12}\right)\left(s+\frac{37}{24}\right)$.

$E_{14}$

s^15 + 22*s^14 + 32417/144*s^13 + 204659/144*s^12 + 342336449/55296*s^11 + 68115593/3456*s^10 + 2263221488671/47775744*s^9 + 4181795833267/47775744*s^8 + 13797972035391089/110075314176*s^7 + 7657276648745269/55037657088*s^6 + 313580047518924277/2641807540224*s^5 + 201947952372690979/2641807540224*s^4 + 82103464370380741201/2282521714753536*s^3 + 6662054508430750633/570630428688384*s^2 + 48009135147253445375/20542695432781824*s + 1116990304605463925/5135673858195456

comment: $f=x^3+y^8+z^2$, with $\mu=14$ and $\operatorname{lct}(f)=\frac{23}{24}$; $b_f(s)=\left(s+\frac{23}{24}\right)(s+1)\left(s+\frac{13}{12}\right)\left(s+\frac{29}{24}\right)\left(s+\frac{31}{24}\right)\left(s+\frac{4}{3}\right)\left(s+\frac{17}{12}\right)\left(s+\frac{35}{24}\right)\left(s+\frac{37}{24}\right)\left(s+\frac{19}{12}\right)\left(s+\frac{5}{3}\right)\left(s+\frac{41}{24}\right)\left(s+\frac{43}{24}\right)\left(s+\frac{23}{12}\right)\left(s+\frac{49}{24}\right)$.

$Z_{11}$

s^12 + 12*s^11 + 2948/45*s^10 + 1936/9*s^9 + 23949178/50625*s^8 + 37153424/50625*s^7 + 1875568552/2278125*s^6 + 510838064/759375*s^5 + 1017959658221/2562890625*s^4 + 423367872884/2562890625*s^3 + 1058990018588/23066015625*s^2 + 176480869264/23066015625*s + 13326080768/23066015625

comment: $f=x^3y+y^5$, with $\mu=11$ and $\operatorname{lct}(f)=\frac{7}{15}$; $b_f(s)=\left(s+\frac{7}{15}\right)\left(s+\frac{2}{3}\right)\left(s+\frac{11}{15}\right)\left(s+\frac{13}{15}\right)\left(s+\frac{14}{15}\right)(s+1)^{2}\left(s+\frac{16}{15}\right)\left(s+\frac{17}{15}\right)\left(s+\frac{19}{15}\right)\left(s+\frac{4}{3}\right)\left(s+\frac{23}{15}\right)$.

$Z_{11}$

s^12 + 35/2*s^11 + 25157/180*s^10 + 242473/360*s^9 + 220863731/101250*s^8 + 4051251127/810000*s^7 + 606847352309/72900000*s^6 + 1477359162433/145800000*s^5 + 5873536085585201/656100000000*s^4 + 7345815518059711/1312200000000*s^3 + 55548474402537101/23619600000000*s^2 + 28151166047923009/47239200000000*s + 1084532078067977/15746400000000

comment: $f=x^3y+y^5+z^2$, with $\mu=11$ and $\operatorname{lct}(f)=\frac{29}{30}$; $b_f(s)=\left(s+\frac{29}{30}\right)(s+1)\left(s+\frac{7}{6}\right)\left(s+\frac{37}{30}\right)\left(s+\frac{41}{30}\right)\left(s+\frac{43}{30}\right)\left(s+\frac{3}{2}\right)\left(s+\frac{47}{30}\right)\left(s+\frac{49}{30}\right)\left(s+\frac{53}{30}\right)\left(s+\frac{11}{6}\right)\left(s+\frac{61}{30}\right)$.

$Z_{12}$

s^12 + 12*s^11 + 720/11*s^10 + 2360/11*s^9 + 626298/1331*s^8 + 968016/1331*s^7 + 1080820/1331*s^6 + 878040/1331*s^5 + 7538287611/19487171*s^4 + 3110169292/19487171*s^3 + 1140021332460/25937424601*s^2 + 188026750800/25937424601*s + 14034384000/25937424601

comment: $f=x^3y+xy^4$, with $\mu=12$ and $\operatorname{lct}(f)=\frac{5}{11}$; $b_f(s)=\left(s+\frac{5}{11}\right)\left(s+\frac{7}{11}\right)\left(s+\frac{8}{11}\right)\left(s+\frac{9}{11}\right)\left(s+\frac{10}{11}\right)(s+1)^{2}\left(s+\frac{12}{11}\right)\left(s+\frac{13}{11}\right)\left(s+\frac{14}{11}\right)\left(s+\frac{15}{11}\right)\left(s+\frac{17}{11}\right)$.

$Z_{12}$

s^12 + 35/2*s^11 + 6147/44*s^10 + 59199/88*s^9 + 5792637/2662*s^8 + 106079253/21296*s^7 + 352386779/42592*s^6 + 855778063/85184*s^5 + 44152347675603/4988715776*s^4 + 5003704192599/907039232*s^3 + 61330358009973645/26559922791424*s^2 + 30957195299119425/53119845582848*s + 3562335017206875/53119845582848

comment: $f=x^3y+xy^4+z^2$, with $\mu=12$ and $\operatorname{lct}(f)=\frac{21}{22}$; $b_f(s)=\left(s+\frac{21}{22}\right)(s+1)\left(s+\frac{25}{22}\right)\left(s+\frac{27}{22}\right)\left(s+\frac{29}{22}\right)\left(s+\frac{31}{22}\right)\left(s+\frac{3}{2}\right)\left(s+\frac{35}{22}\right)\left(s+\frac{37}{22}\right)\left(s+\frac{39}{22}\right)\left(s+\frac{41}{22}\right)\left(s+\frac{45}{22}\right)$.

$Z_{13}$

s^14 + 14*s^13 + 9763/108*s^12 + 3211/9*s^11 + 33641153/34992*s^10 + 32730997/17496*s^9 + 92185834507/34012224*s^8 + 12624448291/4251528*s^7 + 1134761523961/459165024*s^6 + 118961166181/76527504*s^5 + 54089929484773/74384733888*s^4 + 4559912107153/18596183472*s^3 + 254502285224821/4518872583696*s^2 + 8904886017335/1129718145924*s + 143151258250/282429536481

comment: $f=x^3y+y^6$, with $\mu=13$ and $\operatorname{lct}(f)=\frac{4}{9}$; $b_f(s)=\left(s+\frac{4}{9}\right)\left(s+\frac{11}{18}\right)\left(s+\frac{13}{18}\right)\left(s+\frac{7}{9}\right)\left(s+\frac{8}{9}\right)\left(s+\frac{17}{18}\right)(s+1)^{2}\left(s+\frac{19}{18}\right)\left(s+\frac{10}{9}\right)\left(s+\frac{11}{9}\right)\left(s+\frac{23}{18}\right)\left(s+\frac{25}{18}\right)\left(s+\frac{14}{9}\right)$.

$Z_{13}$

s^14 + 41/2*s^13 + 20995/108*s^12 + 244127/216*s^11 + 157486589/34992*s^10 + 908736361/69984*s^9 + 952000912915/34012224*s^8 + 3114951549983/68024448*s^7 + 52467070132529/918330048*s^6 + 24839441843047/459165024*s^5 + 1423101422655575/37192366944*s^4 + 2916381565452407/148769467776*s^3 + 31074411473855539/4518872583696*s^2 + 13360456008577295/9037745167392*s + 27665032520125/188286357654

comment: $f=x^3y+y^6+z^2$, with $\mu=13$ and $\operatorname{lct}(f)=\frac{17}{18}$; $b_f(s)=\left(s+\frac{17}{18}\right)(s+1)\left(s+\frac{10}{9}\right)\left(s+\frac{11}{9}\right)\left(s+\frac{23}{18}\right)\left(s+\frac{25}{18}\right)\left(s+\frac{13}{9}\right)\left(s+\frac{3}{2}\right)\left(s+\frac{14}{9}\right)\left(s+\frac{29}{18}\right)\left(s+\frac{31}{18}\right)\left(s+\frac{16}{9}\right)\left(s+\frac{17}{9}\right)\left(s+\frac{37}{18}\right)$.

$W_{12}$

s^13 + 13*s^12 + 1549/20*s^11 + 5599/20*s^10 + 54787491/80000*s^9 + 95767419/80000*s^8 + 4921167427/3200000*s^7 + 4703112469/3200000*s^6 + 26740985722033/25600000000*s^5 + 2790832682033/5120000000*s^4 + 51943022730987/256000000000*s^3 + 13054371146361/256000000000*s^2 + 1984136005191249/256000000000000*s + 137637748497249/256000000000000

comment: $f=x^4+y^5$, with $\mu=12$ and $\operatorname{lct}(f)=\frac{9}{20}$; $b_f(s)=\left(s+\frac{9}{20}\right)\left(s+\frac{13}{20}\right)\left(s+\frac{7}{10}\right)\left(s+\frac{17}{20}\right)\left(s+\frac{9}{10}\right)\left(s+\frac{19}{20}\right)(s+1)\left(s+\frac{21}{20}\right)\left(s+\frac{11}{10}\right)\left(s+\frac{23}{20}\right)\left(s+\frac{13}{10}\right)\left(s+\frac{27}{20}\right)\left(s+\frac{31}{20}\right)$.

$W_{12}$

s^13 + 19*s^12 + 3319/20*s^11 + 4411/5*s^10 + 254767491/80000*s^9 + 659437383/80000*s^8 + 50363953387/3200000*s^7 + 1436372663/64000*s^6 + 611862491020033/25600000000*s^5 + 480587293252231/25600000000*s^4 + 1352992228614561/128000000000*s^3 + 517124103389829/128000000000*s^2 + 7515321766701957/8000000000000*s + 100361286023229/1000000000000

comment: $f=x^4+y^5+z^2$, with $\mu=12$ and $\operatorname{lct}(f)=\frac{19}{20}$; $b_f(s)=\left(s+\frac{19}{20}\right)(s+1)\left(s+\frac{23}{20}\right)\left(s+\frac{6}{5}\right)\left(s+\frac{27}{20}\right)\left(s+\frac{7}{5}\right)\left(s+\frac{29}{20}\right)\left(s+\frac{31}{20}\right)\left(s+\frac{8}{5}\right)\left(s+\frac{33}{20}\right)\left(s+\frac{9}{5}\right)\left(s+\frac{37}{20}\right)\left(s+\frac{41}{20}\right)$.

$W_{13}$

s^14 + 14*s^13 + 5785/64*s^12 + 5707/16*s^11 + 31486819/32768*s^10 + 30623983/16384*s^9 + 11352970655/4194304*s^8 + 1553844383/524288*s^7 + 10586715690161/4294967296*s^6 + 3326730935827/2147483648*s^5 + 99496629277955/137438953472*s^4 + 8377901299651/34359738368*s^3 + 983849899383225/17592186044416*s^2 + 68735459972025/8796093022208*s + 8822474285625/17592186044416

comment: $f=x^4+xy^4$, with $\mu=13$ and $\operatorname{lct}(f)=\frac{7}{16}$; $b_f(s)=\left(s+\frac{7}{16}\right)\left(s+\frac{5}{8}\right)\left(s+\frac{11}{16}\right)\left(s+\frac{13}{16}\right)\left(s+\frac{7}{8}\right)\left(s+\frac{15}{16}\right)(s+1)^{2}\left(s+\frac{17}{16}\right)\left(s+\frac{9}{8}\right)\left(s+\frac{19}{16}\right)\left(s+\frac{21}{16}\right)\left(s+\frac{11}{8}\right)\left(s+\frac{25}{16}\right)$.

$W_{13}$

s^14 + 41/2*s^13 + 12441/64*s^12 + 144651/128*s^11 + 147442659/32768*s^10 + 850646199/65536*s^9 + 117329094623/4194304*s^8 + 383808775753/8388608*s^7 + 245111109598641/4294967296*s^6 + 464018100896961/8589934592*s^5 + 5249291076745443/137438953472*s^4 + 5376441724368273/274877906944*s^3 + 120645501840410265/17592186044416*s^2 + 51844246845788925/35184372088832*s + 5149899596716875/35184372088832

comment: $f=x^4+xy^4+z^2$, with $\mu=13$ and $\operatorname{lct}(f)=\frac{15}{16}$; $b_f(s)=\left(s+\frac{15}{16}\right)(s+1)\left(s+\frac{9}{8}\right)\left(s+\frac{19}{16}\right)\left(s+\frac{21}{16}\right)\left(s+\frac{11}{8}\right)\left(s+\frac{23}{16}\right)\left(s+\frac{3}{2}\right)\left(s+\frac{25}{16}\right)\left(s+\frac{13}{8}\right)\left(s+\frac{27}{16}\right)\left(s+\frac{29}{16}\right)\left(s+\frac{15}{8}\right)\left(s+\frac{33}{16}\right)$.

$Q_{10}$

s^11 + 16*s^10 + 16675/144*s^9 + 72055/144*s^8 + 237937259/165888*s^7 + 237513367/82944*s^6 + 194097755675/47775744*s^5 + 195683205245/47775744*s^4 + 316508273684081/110075314176*s^3 + 36835459732529/27518828544*s^2 + 368390277514175/990677827584*s + 11565800393525/247669456896

comment: $f=x^3+y^4+yz^2$, with $\mu=10$ and $\operatorname{lct}(f)=\frac{23}{24}$; $b_f(s)=\left(s+\frac{23}{24}\right)(s+1)\left(s+\frac{29}{24}\right)\left(s+\frac{31}{24}\right)\left(s+\frac{4}{3}\right)\left(s+\frac{35}{24}\right)\left(s+\frac{37}{24}\right)\left(s+\frac{5}{3}\right)\left(s+\frac{41}{24}\right)\left(s+\frac{43}{24}\right)\left(s+\frac{49}{24}\right)$.

$Q_{11}$

s^12 + 35/2*s^11 + 3773/27*s^10 + 36355/54*s^9 + 38131643/17496*s^8 + 174755845/34992*s^7 + 35312523301/4251528*s^6 + 42944485045/4251528*s^5 + 10915129697549/1224440064*s^4 + 13633881643855/2448880128*s^3 + 25737110667809/11019960576*s^2 + 13021619252305/22039921152*s + 125181142625/1836660096

comment: $f=x^3+y^2z+xz^3$, with $\mu=11$ and $\operatorname{lct}(f)=\frac{17}{18}$; $b_f(s)=\left(s+\frac{17}{18}\right)(s+1)\left(s+\frac{7}{6}\right)\left(s+\frac{23}{18}\right)\left(s+\frac{4}{3}\right)\left(s+\frac{25}{18}\right)\left(s+\frac{3}{2}\right)\left(s+\frac{29}{18}\right)\left(s+\frac{5}{3}\right)\left(s+\frac{31}{18}\right)\left(s+\frac{11}{6}\right)\left(s+\frac{37}{18}\right)$.

$Q_{12}$

s^11 + 16*s^10 + 5207/45*s^9 + 22466/45*s^8 + 72281923/50625*s^7 + 28772996/10125*s^6 + 9149793433/2278125*s^5 + 9182118226/2278125*s^4 + 7212902790476/2562890625*s^3 + 3337240342904/2562890625*s^2 + 8286704199872/23066015625*s + 206554251904/4613203125

comment: $f=x^3+y^5+yz^2$, with $\mu=12$ and $\operatorname{lct}(f)=\frac{14}{15}$; $b_f(s)=\left(s+\frac{14}{15}\right)(s+1)\left(s+\frac{17}{15}\right)\left(s+\frac{19}{15}\right)\left(s+\frac{4}{3}\right)\left(s+\frac{22}{15}\right)\left(s+\frac{23}{15}\right)\left(s+\frac{5}{3}\right)\left(s+\frac{26}{15}\right)\left(s+\frac{28}{15}\right)\left(s+\frac{31}{15}\right)$.

$S_{11}$

s^12 + 35/2*s^11 + 8943/64*s^10 + 86163/128*s^9 + 71396523/32768*s^8 + 327146589/65536*s^7 + 34812953747/4194304*s^6 + 84649632067/8388608*s^5 + 38236354002033/4294967296*s^4 + 47741409167067/8589934592*s^3 + 160145765639235/68719476736*s^2 + 80983371912975/137438953472*s + 9336777305625/137438953472

comment: $f=x^4+y^2z+xz^2$, with $\mu=11$ and $\operatorname{lct}(f)=\frac{15}{16}$; $b_f(s)=\left(s+\frac{15}{16}\right)(s+1)\left(s+\frac{19}{16}\right)\left(s+\frac{5}{4}\right)\left(s+\frac{21}{16}\right)\left(s+\frac{23}{16}\right)\left(s+\frac{3}{2}\right)\left(s+\frac{25}{16}\right)\left(s+\frac{27}{16}\right)\left(s+\frac{7}{4}\right)\left(s+\frac{29}{16}\right)\left(s+\frac{33}{16}\right)$.

$S_{12}$

s^13 + 19*s^12 + 2157/13*s^11 + 11463/13*s^10 + 6989691/2197*s^9 + 1390773/169*s^8 + 5829195791/371293*s^7 + 8303926349/371293*s^6 + 1492700835492/62748517*s^5 + 1170797428188/62748517*s^4 + 111230094190752/10604499373*s^3 + 42435351828288/10604499373*s^2 + 21633956197470720/23298085122481*s + 2305909870387200/23298085122481

comment: $f=x^2y+y^2z+xz^3$, with $\mu=12$ and $\operatorname{lct}(f)=\frac{12}{13}$; $b_f(s)=\left(s+\frac{12}{13}\right)(s+1)\left(s+\frac{15}{13}\right)\left(s+\frac{16}{13}\right)\left(s+\frac{17}{13}\right)\left(s+\frac{18}{13}\right)\left(s+\frac{19}{13}\right)\left(s+\frac{20}{13}\right)\left(s+\frac{21}{13}\right)\left(s+\frac{22}{13}\right)\left(s+\frac{23}{13}\right)\left(s+\frac{24}{13}\right)\left(s+\frac{27}{13}\right)$.

$U_{12}$

s^10 + 29/2*s^9 + 4511/48*s^8 + 34417/96*s^7 + 6161125/6912*s^6 + 20862437/13824*s^5 + 5261008967/2985984*s^4 + 8363522149/5971968*s^3 + 8662307659/11943936*s^2 + 5277730535/23887872*s + 239383375/7962624

comment: $f=x^3+y^3+z^4$, with $\mu=12$ and $\operatorname{lct}(f)=\frac{11}{12}$; $b_f(s)=\left(s+\frac{11}{12}\right)(s+1)\left(s+\frac{7}{6}\right)\left(s+\frac{5}{4}\right)\left(s+\frac{17}{12}\right)\left(s+\frac{3}{2}\right)\left(s+\frac{19}{12}\right)\left(s+\frac{7}{4}\right)\left(s+\frac{11}{6}\right)\left(s+\frac{25}{12}\right)$.

Definition

$b_f(s)\in\mathbb{Q}[s]$ is the local Bernstein-Sato polynomial [2] at the origin of $f+x_{c+1}^2+\cdots+x_n^2$. Here $f$ is the normal form in $c$ variables $x,y,z$ of a simple singularity, a parabolic unimodal singularity, or an exceptional unimodal singularity in Arnold's classification [1] with the modulus set to $0$ for the unimodal families. The normal forms are $A_k:x^{k+1}$; $D_k:x^2y+y^{k-1}$; $E_6:x^3+y^4$; $E_7:x^3+xy^3$; $E_8:x^3+y^5$; $P_8:x^3+y^3+z^3$; $X_9:x^4+y^4$; $J_{10}:x^3+y^6$; $E_{12}:x^3+y^7$; $E_{13}:x^3+xy^5$; $E_{14}:x^3+y^8$; $Z_{11}:x^3y+y^5$; $Z_{12}:x^3y+xy^4$; $Z_{13}:x^3y+y^6$; $W_{12}:x^4+y^5$; $W_{13}:x^4+xy^4$; $Q_{10}:x^3+y^4+yz^2$; $Q_{11}:x^3+y^2z+xz^3$; $Q_{12}:x^3+y^5+yz^2$; $S_{11}:x^4+y^2z+xz^2$; $S_{12}:x^2y+y^2z+xz^3$; and $U_{12}:x^3+y^3+z^4$.

Parameters

singularity

— singularity ($A_k$ with $k\geq1$, $D_k$ with $k\geq4$, one of $E_6,E_7,E_8$, or one of Arnold's parabolic or exceptional unimodal labels $P_8,X_9,J_{10},E_{12},E_{13},E_{14},Z_{11},Z_{12},Z_{13}, W_{12},W_{13},Q_{10},Q_{11},Q_{12},S_{11},S_{12},U_{12}$)

$n$

— number of variables (the number of variables, at least the corank of the singularity)

Formulas

(1)

If $f$ has weighted degree $1$ with weights $w_i$, and $x^a$ runs through a monomial basis of the Milnor algebra $\mathbb{C}[x_1,\ldots,x_n]/(\partial f)$, then $b_f(s)=(s+1)\prod_{\alpha\in A_f}(s+\alpha)$, where $A_f$ is the set of distinct values $\alpha=\sum_i w_i+\sum_i a_iw_i$.

(2)

$\tilde b_{f+u^2}(s)=\tilde b_f(s+\frac{1}{2})$.

Comments

(3)

The full Bernstein-Sato polynomial is stored, including the factor $s+1$. The reduced Bernstein-Sato polynomial is $\tilde b_f(s)=b_f(s)/(s+1)$ and is not stored separately; when $-1$ is a root of $\tilde b_f(s)$, the full polynomial has $(s+1)^2$ as a factor.

(4)

$\mu$ is the Milnor number $\dim_{\mathbb{C}}\mathbb{C}[x_1,\ldots,x_n]/(\partial f)$, and $\operatorname{lct}(f)$ is the log canonical threshold, which is minus the largest root of $b_f(s)$.

(5)

The polynomial is the local b-function at the origin. For these quasi-homogeneous isolated singularities the origin is the only critical point on $f=0$, so Singular's global routine bfct [3] computes the same polynomial.

(6)

The hyperbolic unimodal families $T_{p,q,r}$ are not included: in the hyperbolic case $1/p+1/q+1/r<1$, so the modulus term $xyz$ is not weighted homogeneous with $x^p,y^q,z^r$. For the exceptional unimodal families the b-function depends on the modulus: $x^3+y^7$ has the root $-32/21$, while $x^3+y^7+xy^5$ has $-11/21$.

Programs

(P1)

Singular

LIB "bfun.lib";
ring r = 0,(x,y),dp;
poly F = x2 + y3;
bfct(F);        // roots -5/6, -1, -7/6

References

[1]

V. I. Arnold, S. M. Gusein-Zade and A. N. Varchenko, Singularities of Differentiable Maps, Volume I, Birkhauser, 1985.

Links

[2]

Wikipedia: Bernstein-Sato polynomial

[3]

Singular manual: bfct

Similar tables

Poincaré polynomials of the finite Coxeter groups — for the ADE rows with $n=3$, the monodromy eigenvalues obtained from the roots of $b_f$ are the eigenvalues of a Coxeter element of the corresponding finite Coxeter group; the degrees read from that Coxeter group's Poincaré polynomial are the Coxeter exponents plus one

Data properties

Entries are of type: rational polynomial

How well the digits are known: exact

Table is complete: no (it holds $A_k$ for $1\leq k\leq12$, $D_k$ for $4\leq k\leq12$, the exceptional simple singularities, and the parabolic and exceptional unimodal singularities in the definition, with $n\leq3$; the hyperbolic unimodal families $T_{p,q,r}$ are not included)

How they were obtained:

The generator computes the weights and a monomial basis of the Milnor algebra over $\mathbb{Q}$, forms the weighted-homogeneous product exactly, and expands it in $\mathbb{Q}[s]$.

Every row was also computed with Singular's bfct [3] and agreed; the control $x^2+y^3$, whose roots are $-5/6,-1,-7/6$, was run first.