restart: with(LinearAlgebra): with(OreModules): march('open',"/Users/aquadrat/Documents/Travail/maple/Personnel/RankFactorizationProblem.mla"):

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SShsaWJuYW1lR0YoNiVRUi9Vc2Vycy9hcXVhZHJhdC9Eb2N1bWVudHMvVHJhdmFpbC9tYXBsZS9QZXJzb25uZWxGKFFWL0xpYnJhcnkvRnJhbWV3b3Jrcy9NYXBsZS5mcmFtZXdvcmsvVmVyc2lvbnMvMjAyMy9saWJGKFFpby9Vc2Vycy9hcXVhZHJhdC9Eb2N1bWVudHMvVHJhdmFpbC9tYXBsZS9QZXJzb25uZWwvUmFua0ZhY3Rvcml6YXRpb25Qcm9ibGVtLm1sYUYoNyVGL0YwRjE=

with(RankFactorizationProblem);

NzZJLUFudGlEaWFnb25hbEc2IkkwQ2VudHJvSGVybWl0aWFuR0YkSS5GYWN0b3JpemF0aW9uR0YkSTVGaW5pdGVGcmVlUmVzb2x1dGlvbkdGJEktRml0dGluZ0lkZWFsR0YkSTJJc0NlbnRyb0hlcm1pdGlhbkdGJEktSXNJbnZlcnRpYmxlR0YkSSxJc05pbHBvdGVudEdGJEkrSXNTb2x1dGlvbkdGJEkqTGVlTWF0cml4R0YkSSlMZWZ0TGlmdEdGJEklTGlmdEdGJEkyTG9jYWxTeXp5Z3lNb2R1bGVHRiRJMlJhbmtGYWN0b3JpemF0aW9uR0YkSTBSZWR1Y2VkU3l6eWdpZXNHRiRJKlJpZ2h0TGlmdEdGJEkrU2F0dXJhdGlvbkdGJEkvU2ltcGxpZmljYXRpb25HRiRJKlNvbHV0aW9uc0dGJEkpU3l6eWdpZXNHRiQ=

R := DefineOreAlgebra(seq(diff=[x[i],t[i]],i=1..4),polynom=[seq(t[i],i=1..4)]): Fitting ideals M := Matrix([[a,b,c],[d,e,f],[g,h,j],[l,m,n]]);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJNR0YoWCwlIUdGKEYoW2dsISIlISEhIy0iJSIkLUkjbWlHNiRGJi9JK21vZHVsZW5hbWVHRihGJDYjUSJhRigtRjE2I1EiZEYoLUYxNiNRImdGKC1GMTYjUSJsRigtRjE2I1EiYkYoLUYxNiNRImVGKC1GMTYjUSJoRigtRjE2I1EibUYoLUYxNiNRImNGKC1GMTYjUSJmRigtRjE2I1EiakYoLUYxNiNRIm5GKDcjLUknTWF0cml4R0YlNiMvSSQlaWRHRigiNTsmNFkkPjopWyQqbyQ=

FittingIdeal(M,0,R);

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

FittingIdeal(M,1,R);

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

FittingIdeal(M,2,R);

Ny5JImFHNiJJImJHRiRJImNHRiRJImRHRiRJImVHRiRJImZHRiRJImdHRiRJImhHRiRJImpHRiRJImxHRiRJIm1HRiRJIm5HRiQ=

FittingIdeal(M,3,R);

NyMiIiI=

FittingIdeal(M,4,R);

NyMiIiI=

M := Matrix([[0,-x[3],0,-x[2]],[0,x[2],0,x[3]],[x[1]+x[4],0,x[1]+x[4],0]]);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJNR0YoWCwlIUdGKEYoW2dsISIlISEhIy0iJCIlLUkjbW5HNiRGJi9JK21vZHVsZW5hbWVHRihGJDYjUSIwRihGMC1JJW1yb3dHRjI2JS1JJW1zdWJHRjI2JC1JI21pR0YyNiNRInhGKC1GMTYjUSIxRigtSSNtb0dGMjYjUScmcGx1cztGKC1GOzYkRj0tRjE2I1EiNEYoLUY4NiQtRkU2I1EqJnVtaW51czA7RigtRjs2JEY9LUYxNiNRIjNGKC1GOzYkRj0tRjE2I1EiMkYoRjBGMEYwRjctRjg2JEZPRldGUkYwNyMtSSdNYXRyaXhHRiU2Iy9JJCVpZEdGKCI1XyVSai5fIilbJCpvJA==

FittingIdeal(M,0,R);

NyMiIiE=

FittingIdeal(M,1,R);

NyUiIiEqJiwmJkkieEc2IjYjIiIiRiomRic2IyIiJUYqRiosJiokJkYnNiMiIiNGMkYqKiQmRic2IyIiJEYyISIiRiosKiomRiZGKkYwRjJGNyomRiZGKkY0RjJGKiomRjBGMkYrRipGNyomRjRGMkYrRipGKg==

FittingIdeal(M,1,R,"reduced");

NyMsKiomJkkieEc2IjYjIiIiIiIiJkkieEc2IjYjIiIjIiIjIiIiKiYmSSJ4RzYiNiMiIiIiIiImSSJ4RzYiNiMiIiQiIiMhIiIqJiZJInhHNiI2IyIiIyIiIyZJInhHNiI2IyIiJSIiIiIiIiomJkkieEc2IjYjIiIkIiIjJkkieEc2IjYjIiIlIiIiISIi

FittingIdeal(M,2,R);

NygiIiEqJiZJInhHNiI2IyIiIyIiIiwmJkYmNiNGKkYqJkYmNiMiIiVGKkYqKiYmRiY2IyIiJEYqRitGKiwkRiQhIiIsJEYxRjYsJiokRiVGKUYqKiRGMkYpRjY=

FittingIdeal(M,2,R,"reduced");

NyUsJiomJkkieEc2IjYjIiIiRikmRiY2IyIiJEYpRikqJkYqRikmRiY2IyIiJUYpRiksJiokJkYmNiMiIiNGNUYpKiRGKkY1ISIiLCYqJkYlRilGM0YpRikqJkYzRilGLkYpRik=

FittingIdeal(M,3,R);

NygiIiEmSSJ4RzYiNiMiIiMmRiU2IyIiJCwkRiQhIiIsJEYpRi0sJiZGJTYjIiIiRjImRiU2IyIiJUYy

FittingIdeal(M,3,R,"reduced");

NyUmSSJ4RzYiNiMiIiQmRiQ2IyIiIywmJkYkNiMiIiJGLiZGJDYjIiIlRi4=

FittingIdeal(M,4,R);

NyMiIiI=

Saturation of an ideal by a polynomial Saturation(x[1]*x[2],[x[1]*x[2]*x[3]],R);

NyMmSSJ4RzYiNiMiIiQ=

Saturation(x[1],[x[2]^2,x[1]*x[3]-x[2]^2],R);

NyQmSSJ4RzYiNiMiIiQqJCZGJDYjIiIjRis=

Saturation(x[3],[x[1]^5*x[3]^3,x[1]*x[2]*x[3],x[2]*x[3]^4],R);

NyQmSSJ4RzYiNiMiIiMqJCZGJDYjIiIiIiIm

Saturation(x[1],[x[1]^2],R);

NyMiIiI=

Saturation(x[1]*x[2],[x[1]^2, x[2]^2],R);

NyMiIiI=

Saturation(x[4],[x[1]*x[2]-x[3]*x[4],x[1]^2*x[3]-x[2]^2*x[4],x[2]^3-x[1]*x[3]^2, x[1]^3-x[2]*x[3]^2],R);

NyksJiomJkkieEc2IjYjIiIiRikmRiY2IyIiI0YpRikqJiZGJjYjIiIkRikmRiY2IyIiJUYpISIiLCYqJkYlRixGLkYpRikqJkYqRixGMUYpRjQsJiomRiVGKUYuRixGNCokRipGMEYpLCYqJEYlRjBGKSomRipGKUYuRixGNCwmKiRGLkYzRikqJkYuRixGMUYsRjQsJiomRipGKUYuRjBGKSooRipGKUYuRilGMUYsRjQsJiomRipGLEYuRixGKSomRipGLEYxRixGNA==

Saturation(0,[x[1]*x[2]],R);

NyMiIiI=

Saturation(x[1]*x[2],[0],R);

NyMiIiE=

Test whether or not an element in a factor polynomial ring is nilpotent IsNilpotent(x[1],Matrix([[x[1]^2]]),R);

SSV0cnVlRyUqcHJvdGVjdGVkRw==

IsNilpotent(x[1]+1,Matrix([[x[1]^2]]),R);

SSZmYWxzZUclKnByb3RlY3RlZEc=

IsNilpotent(x[1],Matrix([[x[1]^3],[x[1]^2*x[2]],[x[1]*x[2]^2],[x[2]^3]]),R);

SSV0cnVlRyUqcHJvdGVjdGVkRw==

IsNilpotent(x[1]*x[2],Matrix([[x[1]^3],[x[1]^2*x[2]],[x[1]*x[2]^2],[x[2]^3]]),R);

SSV0cnVlRyUqcHJvdGVjdGVkRw==

IsNilpotent(x[1]*x[2]+x[1],Matrix([[x[1]^3],[x[1]^2*x[2]],[x[1]*x[2]^2],[x[2]^3]]),R);

SSV0cnVlRyUqcHJvdGVjdGVkRw==

Factorize(Matrix([[(x[1]*x[2]+x[1])^3]]),Matrix([[x[1]^3],[x[1]^2*x[2]],[x[1]*x[2]^2],[x[2]^3]]),R);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyNYLCUhR0YoRihbZ2whIiUhISEjJSIiIiUtSSNtbkc2JEYmL0krbW9kdWxlbmFtZUdGKEYkNiNRIjFGKC1JJW1yb3dHRjA2JS1GLzYjUSIzRigtSSNtb0dGMDYjUTEmSW52aXNpYmxlVGltZXM7RigtSSVtc3ViR0YwNiQtSSNtaUdGMDYjUSJ4RihGLi1GNjYlRjhGOy1JKG1zdWJzdXBHRjA2JUZCRi4tRi82I1EiMkYoLUZJNiVGQkYuRjg3Iy1JJ01hdHJpeEdGJTYjL0kkJWlkR0YoIjVfJW96M18iKVskKm8k

IsNilpotent(x[2],Matrix([[x[1]^3],[x[1]^2*x[2]],[x[1]*x[2]^2],[x[2]^3]]),R);

SSV0cnVlRyUqcHJvdGVjdGVkRw==

IsNilpotent(0,Matrix([[x[1]^3],[x[1]^2*x[2]],[x[1]*x[2]^2],[x[2]^3]]),R);

SSV0cnVlRyUqcHJvdGVjdGVkRw==

Test whether or not an element in a factor polynomial ring is invertible IsInvertible(x[1]+1,Matrix([[x[1]^2]]),R);

SSV0cnVlRyUqcHJvdGVjdGVkRw==

IsInvertible(x[1],Matrix([[x[1]^2]]),R);

SSZmYWxzZUclKnByb3RlY3RlZEc=

IsInvertible(x[1]+1,Matrix([[x[1]^3],[x[1]^2*x[2]],[x[1]*x[2]^2],[x[2]^3]]),R);

SSV0cnVlRyUqcHJvdGVjdGVkRw==

LeftInverse(Matrix([[x[1]+1],[x[1]^3],[x[1]^2*x[2]],[x[1]*x[2]^2],[x[2]^3]]),R);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyNYLCUhR0YoRihbZ2whIiUhISEjJiIiIiYtSSVtcm93RzYkRiYvSSttb2R1bGVuYW1lR0YoRiQ2Jy1JKG1zdWJzdXBHRjA2JS1JI21pR0YwNiNRInhGKC1JI21uR0YwNiNRIjFGKC1GPDYjUSIyRigtSSNtb0dGMDYjUSgmbWludXM7RigtSSVtc3ViR0YwNiRGN0Y7LUZDNiNRJyZwbHVzO0YoRjstRjw2I1ErJnVtaW51czA7MUYoLUY8NiNRIjBGKEZPRk83Iy1JJ01hdHJpeEdGJTYjL0kkJWlkR0YoIjVHKyR6M18iKVskKm8k

Computation of syzygies for a finitely presented module over a factor polynomial ring M := Matrix([[x[1]+1,0],[0,x[1]-1]]);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJNR0YoWCwlIUdGKEYoW2dsISIlISEhIyUiIyIjLUklbXJvd0c2JEYmL0krbW9kdWxlbmFtZUdGKEYkNiUtSSVtc3ViR0YyNiQtSSNtaUdGMjYjUSJ4RigtSSNtbkdGMjYjUSIxRigtSSNtb0dGMjYjUScmcGx1cztGKEY9LUY+NiNRIjBGKEZFLUYxNiVGNi1GQjYjUSgmbWludXM7RihGPTcjLUknTWF0cml4R0YlNiMvSSQlaWRHRigiNUdDJHozXyIpWyQqbyQ=

Syzygies(M,[],R);

LUkkSU5KRzYiNiMiIiM=

Syzygies(M,Matrix([[0]]),R);

LUkkSU5KRzYiNiMiIiM=

K := Syzygies(M,Matrix([[x[1]^2-1]]),R);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJLR0YoWCwlIUdGKEYoW2dsISIlISEhIyUiIyIjLUklbXJvd0c2JEYmL0krbW9kdWxlbmFtZUdGKEYkNiUtSSVtc3ViR0YyNiQtSSNtaUdGMjYjUSJ4RigtSSNtbkdGMjYjUSIxRigtSSNtb0dGMjYjUSgmbWludXM7RihGPS1GPjYjUSIwRihGRS1GMTYlRjYtRkI2I1EnJnBsdXM7RihGPTcjLUknTWF0cml4R0YlNiMvSSQlaWRHRigiNVdnMUw+OilbJCpvJA==

Mult(K,M,R);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyNYLCUhR0YoRihbZ2whIiUhISEjJSIjIiMtSSVtcm93RzYkRiYvSSttb2R1bGVuYW1lR0YoRiQ2JS1JKG1zdWJzdXBHRjA2JS1JI21pR0YwNiNRInhGKC1JI21uR0YwNiNRIjFGKC1GPDYjUSIyRigtSSNtb0dGMDYjUSgmbWludXM7RihGOy1GPDYjUSIwRihGRkYuNyMtSSdNYXRyaXhHRiU2Iy9JJCVpZEdGKCI1KSk0MUw+OilbJCpvJA==

Simplification(%,Matrix([[x[1]^2-1]]),R);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMtSSdSVEFCTEVHRig2JSI1N3EwTD46KVskKm8kLUknTUFUUklYR0YoNiM3JDckLUkjbW5HNiRGJi9JK21vZHVsZW5hbWVHRihGJDYjUSIwRihGNUY0SSdNYXRyaXhHRiU3Iy1GPDYjL0kkJWlkR0YoRi8=

Syzygies(K,Matrix([[x[1]^2-1]]),R);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyNYLCUhR0YoRihbZ2whIiUhISEjJSIjIiMtSSVtcm93RzYkRiYvSSttb2R1bGVuYW1lR0YoRiQ2JS1JJW1zdWJHRjA2JC1JI21pR0YwNiNRInhGKC1JI21uR0YwNiNRIjFGKC1JI21vR0YwNiNRJyZwbHVzO0YoRjstRjw2I1EiMEYoRkMtRi82JUY0LUZANiNRKCZtaW51cztGKEY7NyMtSSdNYXRyaXhHRiU2Iy9JJCVpZEdGKCI1d1cwTD46KVskKm8k

L := Syzygies(K,Matrix([[x[1]^2-1],[x[2]*(x[1]+1)-1]]),R);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJMR0YoLUknUlRBQkxFR0YoNiUiNS9NLUw+OilbJCpvJC1JJ01BVFJJWEdGKDYjNyQ3JC1JI21uRzYkRiYvSSttb2R1bGVuYW1lR0YoRiQ2I1EiMUYoLUY4NiNRIjBGKDckLUY4NiNRIjJGKEY+SSdNYXRyaXhHRiU3Iy1GRTYjL0kkJWlkR0YoRjE=

ReducedSyzygies(K,Matrix([[x[1]^2-1],[x[2]*(x[1]+1)-1]]),R);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMtSSdSVEFCTEVHRig2JSI1WyhRKVE/OilbJCpvJC1JJ01BVFJJWEdGKDYjNyM3JC1JI21uRzYkRiYvSSttb2R1bGVuYW1lR0YoRiQ2I1EiMkYoLUY2NiNRIjBGKEknTWF0cml4R0YlNyMtRj82Iy9JJCVpZEdGKEYv

L := Syzygies(K,Matrix([[x[1]^2-1],[x[2]*(x[1]-1)-1]]),R);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJMR0YoLUknUlRBQkxFR0YoNiUiNVs2JSpRPzopWyQqbyQtSSdNQVRSSVhHRig2IzckNyQtSSNtbkc2JEYmL0krbW9kdWxlbmFtZUdGKEYkNiNRIjBGKC1GODYjUSIxRig3JEY3LUY4NiNRKyZ1bWludXMwOzJGKEknTWF0cml4R0YlNyMtRkU2Iy9JJCVpZEdGKEYx

K2 := Syzygies(M,Matrix([[x[1]^2-1],[x[2]*(x[1]-1)-1]]),R);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNLMkdGKC1JJ1JUQUJMRUdGKDYlIjUzaFk+PzopWyQqbyQtSSdNQVRSSVhHRig2IzckNyQtSSNtbkc2JEYmL0krbW9kdWxlbmFtZUdGKEYkNiNRIjFGKC1GODYjUSIwRig3JC1GODYjUSsmdW1pbnVzMDsyRihGPkknTWF0cml4R0YlNyMtRkU2Iy9JJCVpZEdGKEYx

P2 := Mult(K2,M,R);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNQMkdGKFgsJSFHRihGKFtnbCEiJSEhISMlIiMiIy1JJW1yb3dHNiRGJi9JK21vZHVsZW5hbWVHRihGJDYlLUklbXN1YkdGMjYkLUkjbWlHRjI2I1EieEYoLUkjbW5HRjI2I1EiMUYoLUkjbW9HRjI2I1EnJnBsdXM7RihGPS1GMTYmLUZCNiNRKiZ1bWludXMwO0YoLUYxNiUtRj42I1EiMkYoLUZCNiNRMSZJbnZpc2libGVUaW1lcztGKEY2LUZCNiNRKCZtaW51cztGKEZMLUY+NiNRIjBGKEZVNyMtSSdNYXRyaXhHRiU2Iy9JJCVpZEdGKCI1czBZPj86KVskKm8k

Simplification(%,Matrix([[x[1]^2-1],[x[2]*(x[1]-1)-1]]),R);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMtSSdSVEFCTEVHRig2JSI1Y2pYPj86KVskKm8kLUknTUFUUklYR0YoNiM3JDckLUkjbW5HNiRGJi9JK21vZHVsZW5hbWVHRihGJDYjUSIwRihGNUY0SSdNYXRyaXhHRiU3Iy1GPDYjL0kkJWlkR0YoRi8=

Example 6 of Inria Report 9438 Q := Matrix([[x[1],x[2],0],[x[2],x[1],0]]);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJRR0YoWCwlIUdGKEYoW2dsISIlISEhIyciIyIkLUklbXN1Ykc2JEYmL0krbW9kdWxlbmFtZUdGKEYkNiQtSSNtaUdGMjYjUSJ4RigtSSNtbkdGMjYjUSIxRigtRjE2JEY2LUY7NiNRIjJGKEY+RjAtRjs2I1EiMEYoRkM3Iy1JJ01hdHJpeEdGJTYjL0kkJWlkR0YoIjU7elg+PzopWyQqbyQ=

Q_t := Transpose(Q);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRRX3RHRihYLCUhR0YoRihbZ2whIiUhISEjJyIkIiMtSSVtc3ViRzYkRiYvSSttb2R1bGVuYW1lR0YoRiQ2JC1JI21pR0YyNiNRInhGKC1JI21uR0YyNiNRIjFGKC1GMTYkRjYtRjs2I1EiMkYoLUY7NiNRIjBGKEY+RjBGQzcjLUknTWF0cml4R0YlNiMvSSQlaWRHRigiNWdHWD4/OilbJCpvJA==

J0 := FittingIdeal(Q,0,R,"reduced");

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNKMEdGKDcjIiIhNyNGLg==

J1 := FittingIdeal(Q,1,R,"reduced");

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNKMUdGKDcjLCYqJCZJInhHRig2IyIiIiIiI0Y0KiQmRjI2I0Y1RjUhIiI3I0Yu

J2 := FittingIdeal(Q,2,R,"reduced");

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNKMkdGKDckJkkieEdGKDYjIiIjJkYwNiMiIiI3I0Yu

J3 := FittingIdeal(Q,3,R,"reduced");

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNKM0dGKDcjIiIiNyNGLg==

K0 := Transpose(Syzygies(Q_t,Transpose(convert(J0,Matrix)),R));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNLMEdGKC1JJ1JUQUJMRUdGKDYlIjVLKFIlPj86KVskKm8kLUknTUFUUklYR0YoNiM3JTcjLUkjbW5HNiRGJi9JK21vZHVsZW5hbWVHRihGJDYjUSIwRihGNjcjLUY4NiNRIjFGKEknTWF0cml4R0YlNyMtRkI2Iy9JJCVpZEdGKEYx

K1 := Transpose(Syzygies(Q_t,Transpose(convert(J1,Matrix)),R));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNLMUdGKFgsJSFHRihGKFtnbCEiJSEhISMqIiQiJC1JJW1yb3dHNiRGJi9JK21vZHVsZW5hbWVHRihGJDYkLUkjbW9HRjI2I1EqJnVtaW51czA7RigtSSVtc3ViR0YyNiQtSSNtaUdGMjYjUSJ4RigtSSNtbkdGMjYjUSIyRigtRjs2JEY9LUZCNiNRIjFGKC1GQjYjUSIwRihGRUYwRkpGSkZKRkc3Iy1JJ01hdHJpeEdGJTYjL0kkJWlkR0YoIjV3a1Y+PzopWyQqbyQ=

Simplification(Mult(Q,K1,R),Transpose(convert(J1,Matrix)),R);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMtSSdSVEFCTEVHRig2JSI1Z1lWPj86KVskKm8kLUknTUFUUklYR0YoNiM3JDclLUkjbW5HNiRGJi9JK21vZHVsZW5hbWVHRihGJDYjUSIwRihGNUY1RjRJJ01hdHJpeEdGJTcjLUY8NiMvSSQlaWRHRihGLw==

K2 := Transpose(Syzygies(Q_t,Transpose(convert(J2,Matrix)),R));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNLMkdGKC1JJ1JUQUJMRUdGKDYlIjVbTFU+PzopWyQqbyQtSSdNQVRSSVhHRig2IzclNyUtSSNtbkc2JEYmL0krbW9kdWxlbmFtZUdGKEYkNiNRIjFGKC1GODYjUSIwRihGPjclRj5GN0Y+NyVGPkY+RjdJJ01hdHJpeEdGJTcjLUZDNiMvSSQlaWRHRihGMQ==

Example 7 of Inria Report 9438 Q := Matrix([[x[1],x[2],2*x[1]+x[2]],[x[2],x[1],x[1]+2*x[2]]]);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJRR0YoWCwlIUdGKEYoW2dsISIlISEhIyciIyIkLUklbXN1Ykc2JEYmL0krbW9kdWxlbmFtZUdGKEYkNiQtSSNtaUdGMjYjUSJ4RigtSSNtbkdGMjYjUSIxRigtRjE2JEY2LUY7NiNRIjJGKEY+RjAtSSVtcm93R0YyNiUtRkQ2JUZALUkjbW9HRjI2I1ExJkludmlzaWJsZVRpbWVzO0YoRjAtRkk2I1EnJnBsdXM7RihGPi1GRDYlRjBGTC1GRDYlRkBGSEY+NyMtSSdNYXRyaXhHRiU2Iy9JJCVpZEdGKCI1OyY0WSQ+OilbJCpvJA==

Q_t := Transpose(Q);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRRX3RHRihYLCUhR0YoRihbZ2whIiUhISEjJyIkIiMtSSVtc3ViRzYkRiYvSSttb2R1bGVuYW1lR0YoRiQ2JC1JI21pR0YyNiNRInhGKC1JI21uR0YyNiNRIjFGKC1GMTYkRjYtRjs2I1EiMkYoLUklbXJvd0dGMjYlLUZENiVGQC1JI21vR0YyNiNRMSZJbnZpc2libGVUaW1lcztGKEYwLUZJNiNRJyZwbHVzO0YoRj5GPkYwLUZENiVGMEZMLUZENiVGQEZIRj43Iy1JJ01hdHJpeEdGJTYjL0kkJWlkR0YoIjU/Z2dNPjopWyQqbyQ=

J0 := FittingIdeal(Q,0,R,"reduced");

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNKMEdGKDcjIiIhNyNGLg==

J1 := FittingIdeal(Q,1,R,"reduced");

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNKMUdGKDcjLCYqJCZJInhHRig2IyIiIiIiI0Y0KiQmRjI2I0Y1RjUhIiI3I0Yu

J2 := FittingIdeal(Q,2,R,"reduced");

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNKMkdGKDckJkkieEdGKDYjIiIjJkYwNiMiIiI3I0Yu

J3 := FittingIdeal(Q,3,R,"reduced");

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNKM0dGKDcjIiIiNyNGLg==

K0 := Transpose(Syzygies(Q_t,Transpose(convert(J0,Matrix)),R));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNLMEdGKC1JJ1JUQUJMRUdGKDYlIjU7cmVNPjopWyQqbyQtSSdNQVRSSVhHRig2IzclNyMtSSNtbkc2JEYmL0krbW9kdWxlbmFtZUdGKEYkNiNRIjJGKDcjLUY4NiNRIjFGKDcjLUY4NiNRKyZ1bWludXMwOzFGKEknTWF0cml4R0YlNyMtRkY2Iy9JJCVpZEdGKEYx

K1 := Transpose(Syzygies(Q_t,Transpose(convert(J1,Matrix)),R));

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

K2 := Transpose(Syzygies(Q_t,Transpose(convert(J2,Matrix)),R));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNLMkdGKC1JJ1JUQUJMRUdGKDYlIjUpeXZYJD46KVskKm8kLUknTUFUUklYR0YoNiM3JTclLUkjbW5HNiRGJi9JK21vZHVsZW5hbWVHRihGJDYjUSIxRigtRjg2I1EiMEYoRj43JUY+RjdGPjclRj5GPkY3SSdNYXRyaXhHRiU3Iy1GQzYjL0kkJWlkR0YoRjE=

Computation of left/right/generalized inverses for a finitely presented module over a factor polynomial ring LeftInverse(Q_t,R);

NyI=

S := LeftLift(Q_t,Matrix([[x[4]*J1[1]-1]]),R);

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

simplify(S.Q_t);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyNYLCUhR0YoRihbZ2whIiUhISEjJSIjIiMtSSVtcm93RzYkRiYvSSttb2R1bGVuYW1lR0YoRiQ2JS1JKG1mZW5jZWRHRjA2Iy1GLzYlLUkobXN1YnN1cEdGMDYlLUkjbWlHRjA2I1EieEYoLUkjbW5HRjA2I1EiMUYoLUZBNiNRIjJGKC1JI21vR0YwNiNRKCZtaW51cztGKC1GOjYlRjxGREZELUZINiNRMSZJbnZpc2libGVUaW1lcztGKC1JJW1zdWJHRjA2JEY8LUZBNiNRIjRGKC1GQTYjUSIwRihGVkYuNyMtSSdNYXRyaXhHRiU2Iy9JJCVpZEdGKCI1Jz4zWSQ+OilbJCpvJA==

Simplification(S.Q_t,Matrix([[x[4]*J1[1]-1]]),R);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMtSSdSVEFCTEVHRig2JSI1b11mTT46KVskKm8kLUknTUFUUklYR0YoNiM3JDckLUkjbW5HNiRGJi9JK21vZHVsZW5hbWVHRihGJDYjUSIxRigtRjY2I1EiMEYoNyRGPEY1SSdNYXRyaXhHRiU3Iy1GQDYjL0kkJWlkR0YoRi8=

T := RightLift(Q,Matrix([[x[4]*J1[1]-1]]),R);

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

Simplification(Q.T,Matrix([[x[4]*J1[1]-1]]),R);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMtSSdSVEFCTEVHRig2JSI1U2JlTT46KVskKm8kLUknTUFUUklYR0YoNiM3JDckLUkjbW5HNiRGJi9JK21vZHVsZW5hbWVHRihGJDYjUSIxRigtRjY2I1EiMEYoNyRGPEY1SSdNYXRyaXhHRiU3Iy1GQDYjL0kkJWlkR0YoRi8=

LeftLift(Q_t,Matrix([[x[4]*J2[1]-1]]),R);

NyI=

LeftLift(Q_t,Matrix([[x[4]*J2[2]-1]]),R);

NyI=

M := Matrix([[x[1]],[x[2]],[x[3]]]);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJNR0YoWCwlIUdGKEYoW2dsISIlISEhIyQiJCIiLUklbXN1Ykc2JEYmL0krbW9kdWxlbmFtZUdGKEYkNiQtSSNtaUdGMjYjUSJ4RigtSSNtbkdGMjYjUSIxRigtRjE2JEY2LUY7NiNRIjJGKC1GMTYkRjYtRjs2I1EiM0YoNyMtSSdNYXRyaXhHRiU2Iy9JJCVpZEdGKCI1L1draEA6KVskKm8k

LeftLift(M,Matrix([[x[4]*x[1]-1]]),R);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyNYLCUhR0YoRihbZ2whIiUhISEjJCIiIiQtSSVtc3ViRzYkRiYvSSttb2R1bGVuYW1lR0YoRiQ2JC1JI21pR0YwNiNRInhGKC1JI21uR0YwNiNRIjRGKC1GOTYjUSIwRihGPDcjLUknTWF0cml4R0YlNiMvSSQlaWRHRigiNXNYaE0+OilbJCpvJA==

LeftLift(M,Matrix([[x[4]*x[2]-1]]),R);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyNYLCUhR0YoRihbZ2whIiUhISEjJCIiIiQtSSNtbkc2JEYmL0krbW9kdWxlbmFtZUdGKEYkNiNRIjBGKC1JJW1zdWJHRjA2JC1JI21pR0YwNiNRInhGKC1GLzYjUSI0RihGLjcjLUknTWF0cml4R0YlNiMvSSQlaWRHRigiNVNQZ00+OilbJCpvJA==

LeftLift(M,Matrix([[x[4]*x[3]-1]]),R);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyNYLCUhR0YoRihbZ2whIiUhISEjJCIiIiQtSSNtbkc2JEYmL0krbW9kdWxlbmFtZUdGKEYkNiNRIjBGKEYuLUklbXN1YkdGMDYkLUkjbWlHRjA2I1EieEYoLUYvNiNRIjRGKDcjLUknTWF0cml4R0YlNiMvSSQlaWRHRigiNVctZ00+OilbJCpvJA==

N := SyzygyModule(M,R);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJOR0YoWCwlIUdGKEYoW2dsISIlISEhIyoiJCIkLUklbXJvd0c2JEYmL0krbW9kdWxlbmFtZUdGKEYkNiQtSSNtb0dGMjYjUSomdW1pbnVzMDtGKC1JJW1zdWJHRjI2JC1JI21pR0YyNiNRInhGKC1JI21uR0YyNiNRIjNGKC1GMTYkRjYtRjs2JEY9LUZCNiNRIjJGKC1GQjYjUSIwRihGTC1GOzYkRj0tRkI2I1EiMUYoRjBGT0ZMRkc3Iy1JJ01hdHJpeEdGJTYjL0kkJWlkR0YoIjVDRGdNPjopWyQqbyQ=

LeftLift(N,Matrix([[x[4]*x[1]-1]]),R);

NyI=

RightLift(N,Matrix([[x[4]*x[1]-1]]),R);

NyI=

P := Lift(N,Matrix([[x[4]*x[1]-1]]),R);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJQR0YoWCwlIUdGKEYoW2dsISIlISEhIyoiJCIkLUkjbW5HNiRGJi9JK21vZHVsZW5hbWVHRihGJDYjUSIwRihGMC1JJW1zdWJHRjI2JC1JI21pR0YyNiNRInhGKC1GMTYjUSI0RihGMEY3RjBGMEYwRjA3Iy1JJ01hdHJpeEdGJTYjL0kkJWlkR0YoIjVnLSxtQDopWyQqbyQ=

Simplification(N.P.N-N,Matrix([[x[4]*x[1]-1]]),R);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMtSSdSVEFCTEVHRig2JSI1N18sbUA6KVskKm8kLUknTUFUUklYR0YoNiM3JTclLUkjbW5HNiRGJi9JK21vZHVsZW5hbWVHRihGJDYjUSIwRihGNUY1RjRGNEknTWF0cml4R0YlNyMtRjw2Iy9JJCVpZEdGKEYv

P2 := Lift(N,Matrix([[x[4]*x[2]-1]]),R);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNQMkdGKFgsJSFHRihGKFtnbCEiJSEhISMqIiQiJC1JI21uRzYkRiYvSSttb2R1bGVuYW1lR0YoRiQ2I1EiMEYoRjBGMC1JJW1yb3dHRjI2JC1JI21vR0YyNiNRKiZ1bWludXMwO0YoLUklbXN1YkdGMjYkLUkjbWlHRjI2I1EieEYoLUYxNiNRIjRGKEYwRjBGMEYwRj43Iy1JJ01hdHJpeEdGJTYjL0kkJWlkR0YoIjVPXWFtQDopWyQqbyQ=

Simplification(N.P2.N-N,Matrix([[x[4]*x[2]-1]]),R);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMtSSdSVEFCTEVHRig2JSI1cypcbDtfIilbJCpvJC1JJ01BVFJJWEdGKDYjNyU3JS1JI21uRzYkRiYvSSttb2R1bGVuYW1lR0YoRiQ2I1EiMEYoRjVGNUY0RjRJJ01hdHJpeEdGJTcjLUY8NiMvSSQlaWRHRihGLw==

P3 := Lift(N,Matrix([[x[4]*x[3]-1]]),R);

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Simplification(N.P3.N-N,Matrix([[x[4]*x[3]-1]]),R);

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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LVEifkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDLyUrZXhlY3V0YWJsZUdGNEYv LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LVEifkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDLyUrZXhlY3V0YWJsZUdGNEYv LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LVEifkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDLyUrZXhlY3V0YWJsZUdGNEYv LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LVEifkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDLyUrZXhlY3V0YWJsZUdGNEYv LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LVEifkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDLyUrZXhlY3V0YWJsZUdGNEYv LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LVEifkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDLyUrZXhlY3V0YWJsZUdGNEYv LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LVEifkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDLyUrZXhlY3V0YWJsZUdGNEYv