Re: [math-fun] computer algebra
Victor Miller> Veit, I gave this to SAGE (which actually uses the Groeber Basis stuff
in SINGULAR) and it fairly quickly calculated a Groebner Basis, and showed that the dimension of the ideal is 3.
I.e., triply underdetermined? That scotches the plan to PSLQ the exact algebraics from 1000 digit approximations. --rwg
Right now I'm waiting for it to produce a primary decomposition which should shed some light on the matter.
Victor
On Wed, Aug 24, 2011 at 1:56 AM, Bill Gosper <billgosper@gmail.com> wrote:
Victor Miller> Veit, I gave this to SAGE (which actually uses the Groeber Basis stuff
in SINGULAR) and it fairly quickly calculated a Groebner Basis, and showed that the dimension of the ideal is 3.
I.e., triply underdetermined?
Yep, That's what it means.
That scotches the plan to PSLQ the exact algebraics from 1000 digit approximations. --rwg
Right now I'm waiting for it to produce a primary decomposition which should shed some light on the matter.
Victor
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Suppose I told you that by further restricting the variables to lie on the unit circle, then the solutions would be isolated. Would that help? These solutions would fill a gap in the classification of order 7 complex Hadamard matrices. A complex Hadamard matrix is a unitary matrix with equal modulus entries. They are usually rescaled so the entries have modulus 1. A complex Hadamard matrix that happens to be real is the more familiar Hadamard matrix, which can exist only when the order is 1, 2, or a multiple of 4. All the known complex Hadamard matrices of order less than 11 have the property that their entries are constructed by taking products of integer powers of a small set of generators of modulus 1. Something strange happens at order 11, where it appears that there are thousands of isolated complex Hadamard matrices without this structure. I've computed some of these to over 100 digits and looked for integer relations among the phase angles using LLL and found none -- an approach to which all the lower order matrices easily yield. Does this elevate your respect for the number 11? Veit On Aug 24, 2011, at 8:19 AM, Victor Miller wrote:
On Wed, Aug 24, 2011 at 1:56 AM, Bill Gosper <billgosper@gmail.com> wrote:
Victor Miller> Veit, I gave this to SAGE (which actually uses the Groeber Basis stuff
in SINGULAR) and it fairly quickly calculated a Groebner Basis, and showed that the dimension of the ideal is 3.
I.e., triply underdetermined?
Yep, That's what it means.
That scotches the plan to PSLQ the exact algebraics from 1000 digit approximations. --rwg
Right now I'm waiting for it to produce a primary decomposition which should shed some light on the matter.
Victor
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Just add three more equations: Try b=c=d=1 first; if that's singular, try things like b=.6 or b+c=1.5. Rich ________________________________________ From: math-fun-bounces@mailman.xmission.com [math-fun-bounces@mailman.xmission.com] on behalf of Bill Gosper [billgosper@gmail.com] Sent: Tuesday, August 23, 2011 11:56 PM To: math-fun@mailman.xmission.com Subject: Re: [math-fun] computer algebra Victor Miller> Veit, I gave this to SAGE (which actually uses the Groeber Basis stuff
in SINGULAR) and it fairly quickly calculated a Groebner Basis, and showed that the dimension of the ideal is 3.
I.e., triply underdetermined? That scotches the plan to PSLQ the exact algebraics from 1000 digit approximations. --rwg
Right now I'm waiting for it to produce a primary decomposition which should shed some light on the matter.
Victor
math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
When you set b=1 SAGE (actually Singular) almost immediately calculates the primary decomposition: [Ideal (f, e, b - 1) of Multivariate Polynomial Ring in b, c, d, e, f over Rational Field, Ideal (f, c, b - 1) of Multivariate Polynomial Ring in b, c, d, e, f over Rational Field, Ideal (e, c, b - 1) of Multivariate Polynomial Ring in b, c, d, e, f over Rational Field, Ideal (f - 1, e - 1, d - 1, b - 1, c^2 + 5*c + 1) of Multivariate Polynomial Ring in b, c, d, e, f over Rational Field, Ideal (f - 1, d - 1, c - 1, b - 1, e^2 + 5*e + 1) of Multivariate Polynomial Ring in b, c, d, e, f over Rational Field, Ideal (e - 1, d - 1, c - 1, b - 1, f^2 + 5*f + 1) of Multivariate Polynomial Ring in b, c, d, e, f over Rational Field, Ideal (e, d, b - 1) of Multivariate Polynomial Ring in b, c, d, e, f over Rational Field, Ideal (f - 1, e - 1, c - 1, b - 1, d^2 + 5*d + 1) of Multivariate Polynomial Ring in b, c, d, e, f over Rational Field, Ideal (f, d, b - 1) of Multivariate Polynomial Ring in b, c, d, e, f over Rational Field, Ideal (d, c, b - 1) of Multivariate Polynomial Ring in b, c, d, e, f over Rational Field] I didn't put in equations to forbid b,c,d,e,f from being 0. That seems to be the source of the higher dimensionality. A typical non-spurious component is b=1,d=1,e=1,f=1, c^2 + 5*c + 1 = 0. On Wed, Aug 24, 2011 at 11:23 AM, Schroeppel, Richard <rschroe@sandia.gov> wrote:
Just add three more equations: Try b=c=d=1 first; if that's singular, try things like b=.6 or b+c=1.5.
Rich ________________________________________ From: math-fun-bounces@mailman.xmission.com [math-fun-bounces@mailman.xmission.com] on behalf of Bill Gosper [billgosper@gmail.com] Sent: Tuesday, August 23, 2011 11:56 PM To: math-fun@mailman.xmission.com Subject: Re: [math-fun] computer algebra
Victor Miller> Veit, I gave this to SAGE (which actually uses the Groeber Basis stuff
in SINGULAR) and it fairly quickly calculated a Groebner Basis, and showed that the dimension of the ideal is 3.
I.e., triply underdetermined?
That scotches the plan to PSLQ the exact algebraics from 1000 digit approximations. --rwg
Right now I'm waiting for it to produce a primary decomposition which should shed some light on the matter.
Victor
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Indeed when I added equations like b*bb-1 == 0, to forbid b from being zero, the resulting ideal is 0 dimensional (which means that there are only a finite number of solutions over C). The groebner basis (in lex order) has 225 elements. Right now I'm attempting to calculate the radical and primary decomposition. Victor On Wed, Aug 24, 2011 at 12:09 PM, Victor Miller <victorsmiller@gmail.com> wrote:
When you set b=1 SAGE (actually Singular) almost immediately calculates the primary decomposition:
[Ideal (f, e, b - 1) of Multivariate Polynomial Ring in b, c, d, e, f over Rational Field, Ideal (f, c, b - 1) of Multivariate Polynomial Ring in b, c, d, e, f over Rational Field, Ideal (e, c, b - 1) of Multivariate Polynomial Ring in b, c, d, e, f over Rational Field, Ideal (f - 1, e - 1, d - 1, b - 1, c^2 + 5*c + 1) of Multivariate Polynomial Ring in b, c, d, e, f over Rational Field, Ideal (f - 1, d - 1, c - 1, b - 1, e^2 + 5*e + 1) of Multivariate Polynomial Ring in b, c, d, e, f over Rational Field, Ideal (e - 1, d - 1, c - 1, b - 1, f^2 + 5*f + 1) of Multivariate Polynomial Ring in b, c, d, e, f over Rational Field, Ideal (e, d, b - 1) of Multivariate Polynomial Ring in b, c, d, e, f over Rational Field, Ideal (f - 1, e - 1, c - 1, b - 1, d^2 + 5*d + 1) of Multivariate Polynomial Ring in b, c, d, e, f over Rational Field, Ideal (f, d, b - 1) of Multivariate Polynomial Ring in b, c, d, e, f over Rational Field, Ideal (d, c, b - 1) of Multivariate Polynomial Ring in b, c, d, e, f over Rational Field]
I didn't put in equations to forbid b,c,d,e,f from being 0. That seems to be the source of the higher dimensionality. A typical non-spurious component is b=1,d=1,e=1,f=1, c^2 + 5*c + 1 = 0.
On Wed, Aug 24, 2011 at 11:23 AM, Schroeppel, Richard <rschroe@sandia.gov> wrote:
Just add three more equations: Try b=c=d=1 first; if that's singular, try things like b=.6 or b+c=1.5.
Rich ________________________________________ From: math-fun-bounces@mailman.xmission.com [math-fun-bounces@mailman.xmission.com] on behalf of Bill Gosper [billgosper@gmail.com] Sent: Tuesday, August 23, 2011 11:56 PM To: math-fun@mailman.xmission.com Subject: Re: [math-fun] computer algebra
Victor Miller> Veit, I gave this to SAGE (which actually uses the Groeber Basis stuff
in SINGULAR) and it fairly quickly calculated a Groebner Basis, and showed that the dimension of the ideal is 3.
I.e., triply underdetermined?
That scotches the plan to PSLQ the exact algebraics from 1000 digit approximations. --rwg
Right now I'm waiting for it to produce a primary decomposition which should shed some light on the matter.
Victor
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Here's the groebner basis (in lex order) for the radical of the ideal. So far I can't persuade SAGE to compute the primary decomposition. If you notice the last element is a polynomial in b alone: Ideal (f*ff - 1, e*ee - 1, d*dd - 1, c*cc - 1, b*bb - 1, b*c*d*e*f^2 + b^2*c*d*f + b*c^2*e*f + b*c*d*e*f + b*d^2*e*f + c*d*e^2*f + b*c*d*e, b*c*d*e^2*f + b^2*c*d*e + b*c*d^2*f + b*c*d*e*f + c^2*d*e*f + b*d*e*f^2 + b*c*e*f, b*c*d^2*e*f + b*c^2*d*e + b^2*d*e*f + b*c*d*e*f + b*c*e^2*f + c*d*e*f^2 + b*c*d*f, b*c^2*d*e*f + b*c*d^2*e + b^2*c*e*f + b*c*d*e*f + b*d*e^2*f + b*c*d*f^2 + c*d*e*f, b^2*c*d*e*f + b*c*d*e^2 + b*c^2*d*f + b*c*d*e*f + c*d^2*e*f + b*c*e*f^2 + b*d*e*f, 400*ff^41 + 3120*ff^40 + 22888*ff^39 + 108912*ff^38 + 99688*ff^37 - 628842*ff^36 - 4051895*ff^35 - 10171717*ff^34 - 8374137*ff^33 + 15303862*ff^32 + 73160225*ff^31 + 184348225*ff^30 + 356022965*ff^29 + 579214135*ff^28 + 831735578*ff^27 + 1054268600*ff^26 + 1186394061*ff^25 + 1185420009*ff^24 + 1017272898*ff^23 + 690503460*ff^22 + 246402021*ff^21 - 246402021*ff^20 - 690503460*ff^19 - 1017272898*ff^18 - 1185420009*ff^17 - 1186394061*ff^16 - 1054268600*ff^15 - 831735578*ff^14 - 579214135*ff^13 - 356022965*ff^12 - 184348225*ff^11 - 73160225*ff^10 - 15303862*ff^9 + 8374137*ff^8 + 10171717*ff^7 + 4051895*ff^6 + 628842*ff^5 - 99688*ff^4 - 108912*ff^3 - 22888*ff^2 - 3120*ff - 400, 400*f^41 + 3120*f^40 + 22888*f^39 + 108912*f^38 + 99688*f^37 - 628842*f^36 - 4051895*f^35 - 10171717*f^34 - 8374137*f^33 + 15303862*f^32 + 73160225*f^31 + 184348225*f^30 + 356022965*f^29 + 579214135*f^28 + 831735578*f^27 + 1054268600*f^26 + 1186394061*f^25 + 1185420009*f^24 + 1017272898*f^23 + 690503460*f^22 + 246402021*f^21 - 246402021*f^20 - 690503460*f^19 - 1017272898*f^18 - 1185420009*f^17 - 1186394061*f^16 - 1054268600*f^15 - 831735578*f^14 - 579214135*f^13 - 356022965*f^12 - 184348225*f^11 - 73160225*f^10 - 15303862*f^9 + 8374137*f^8 + 10171717*f^7 + 4051895*f^6 + 628842*f^5 - 99688*f^4 - 108912*f^3 - 22888*f^2 - 3120*f - 400, 1152*ee^64 + 16704*ee^63 + 148320*ee^62 + 903792*ee^61 + 4770080*ee^60 + 23387824*ee^59 + 92591288*ee^58 + 331368468*ee^57 + 1057959312*ee^56 + 2878780540*ee^55 + 6267271526*ee^54 + 8995623231*ee^53 - 4736672226*ee^52 - 99081529844*ee^51 - 473493845753*ee^50 - 1638775838283*ee^49 - 4721423149982*ee^48 - 11895183084400*ee^47 - 26805051366282*ee^46 - 54609059293775*ee^45 - 101147125314256*ee^44 - 171048948174825*ee^43 - 265284635327558*ee^42 - 379155055292650*ee^41 - 501563808003370*ee^40 - 615845813553397*ee^39 - 702041970016373*ee^38 - 740376567640206*ee^37 - 715427568687790*ee^36 - 620027326148923*ee^35 - 457749082379246*ee^34 - 243203019682098*ee^33 + 243203019682098*ee^31 + 457749082379246*ee^30 + 620027326148923*ee^29 + 715427568687790*ee^28 + 740376567640206*ee^27 + 702041970016373*ee^26 + 615845813553397*ee^25 + 501563808003370*ee^24 + 379155055292650*ee^23 + 265284635327558*ee^22 + 171048948174825*ee^21 + 101147125314256*ee^20 + 54609059293775*ee^19 + 26805051366282*ee^18 + 11895183084400*ee^17 + 4721423149982*ee^16 + 1638775838283*ee^15 + 473493845753*ee^14 + 99081529844*ee^13 + 4736672226*ee^12 - 8995623231*ee^11 - 6267271526*ee^10 - 2878780540*ee^9 - 1057959312*ee^8 - 331368468*ee^7 - 92591288*ee^6 - 23387824*ee^5 - 4770080*ee^4 - 903792*ee^3 - 148320*ee^2 - 16704*ee - 1152, 1152*dd^64 + 16704*dd^63 + 148320*dd^62 + 903792*dd^61 + 4770080*dd^60 + 23387824*dd^59 + 92591288*dd^58 + 331368468*dd^57 + 1057959312*dd^56 + 2878780540*dd^55 + 6267271526*dd^54 + 8995623231*dd^53 - 4736672226*dd^52 - 99081529844*dd^51 - 473493845753*dd^50 - 1638775838283*dd^49 - 4721423149982*dd^48 - 11895183084400*dd^47 - 26805051366282*dd^46 - 54609059293775*dd^45 - 101147125314256*dd^44 - 171048948174825*dd^43 - 265284635327558*dd^42 - 379155055292650*dd^41 - 501563808003370*dd^40 - 615845813553397*dd^39 - 702041970016373*dd^38 - 740376567640206*dd^37 - 715427568687790*dd^36 - 620027326148923*dd^35 - 457749082379246*dd^34 - 243203019682098*dd^33 + 243203019682098*dd^31 + 457749082379246*dd^30 + 620027326148923*dd^29 + 715427568687790*dd^28 + 740376567640206*dd^27 + 702041970016373*dd^26 + 615845813553397*dd^25 + 501563808003370*dd^24 + 379155055292650*dd^23 + 265284635327558*dd^22 + 171048948174825*dd^21 + 101147125314256*dd^20 + 54609059293775*dd^19 + 26805051366282*dd^18 + 11895183084400*dd^17 + 4721423149982*dd^16 + 1638775838283*dd^15 + 473493845753*dd^14 + 99081529844*dd^13 + 4736672226*dd^12 - 8995623231*dd^11 - 6267271526*dd^10 - 2878780540*dd^9 - 1057959312*dd^8 - 331368468*dd^7 - 92591288*dd^6 - 23387824*dd^5 - 4770080*dd^4 - 903792*dd^3 - 148320*dd^2 - 16704*dd - 1152, 1152*e^64 + 16704*e^63 + 148320*e^62 + 903792*e^61 + 4770080*e^60 + 23387824*e^59 + 92591288*e^58 + 331368468*e^57 + 1057959312*e^56 + 2878780540*e^55 + 6267271526*e^54 + 8995623231*e^53 - 4736672226*e^52 - 99081529844*e^51 - 473493845753*e^50 - 1638775838283*e^49 - 4721423149982*e^48 - 11895183084400*e^47 - 26805051366282*e^46 - 54609059293775*e^45 - 101147125314256*e^44 - 171048948174825*e^43 - 265284635327558*e^42 - 379155055292650*e^41 - 501563808003370*e^40 - 615845813553397*e^39 - 702041970016373*e^38 - 740376567640206*e^37 - 715427568687790*e^36 - 620027326148923*e^35 - 457749082379246*e^34 - 243203019682098*e^33 + 243203019682098*e^31 + 457749082379246*e^30 + 620027326148923*e^29 + 715427568687790*e^28 + 740376567640206*e^27 + 702041970016373*e^26 + 615845813553397*e^25 + 501563808003370*e^24 + 379155055292650*e^23 + 265284635327558*e^22 + 171048948174825*e^21 + 101147125314256*e^20 + 54609059293775*e^19 + 26805051366282*e^18 + 11895183084400*e^17 + 4721423149982*e^16 + 1638775838283*e^15 + 473493845753*e^14 + 99081529844*e^13 + 4736672226*e^12 - 8995623231*e^11 - 6267271526*e^10 - 2878780540*e^9 - 1057959312*e^8 - 331368468*e^7 - 92591288*e^6 - 23387824*e^5 - 4770080*e^4 - 903792*e^3 - 148320*e^2 - 16704*e - 1152, 1152*d^64 + 16704*d^63 + 148320*d^62 + 903792*d^61 + 4770080*d^60 + 23387824*d^59 + 92591288*d^58 + 331368468*d^57 + 1057959312*d^56 + 2878780540*d^55 + 6267271526*d^54 + 8995623231*d^53 - 4736672226*d^52 - 99081529844*d^51 - 473493845753*d^50 - 1638775838283*d^49 - 4721423149982*d^48 - 11895183084400*d^47 - 26805051366282*d^46 - 54609059293775*d^45 - 101147125314256*d^44 - 171048948174825*d^43 - 265284635327558*d^42 - 379155055292650*d^41 - 501563808003370*d^40 - 615845813553397*d^39 - 702041970016373*d^38 - 740376567640206*d^37 - 715427568687790*d^36 - 620027326148923*d^35 - 457749082379246*d^34 - 243203019682098*d^33 + 243203019682098*d^31 + 457749082379246*d^30 + 620027326148923*d^29 + 715427568687790*d^28 + 740376567640206*d^27 + 702041970016373*d^26 + 615845813553397*d^25 + 501563808003370*d^24 + 379155055292650*d^23 + 265284635327558*d^22 + 171048948174825*d^21 + 101147125314256*d^20 + 54609059293775*d^19 + 26805051366282*d^18 + 11895183084400*d^17 + 4721423149982*d^16 + 1638775838283*d^15 + 473493845753*d^14 + 99081529844*d^13 + 4736672226*d^12 - 8995623231*d^11 - 6267271526*d^10 - 2878780540*d^9 - 1057959312*d^8 - 331368468*d^7 - 92591288*d^6 - 23387824*d^5 - 4770080*d^4 - 903792*d^3 - 148320*d^2 - 16704*d - 1152, 1843200*cc^69 + 20828160*cc^68 + 105560064*cc^67 + 312038400*cc^66 + 593355776*cc^65 + 1189639680*cc^64 - 1962641664*cc^63 - 27145893376*cc^62 - 79472117632*cc^61 - 141495192896*cc^60 - 318179768512*cc^59 - 558887133344*cc^58 - 1540921488048*cc^57 - 3681309755424*cc^56 - 7606453336976*cc^55 - 14600554466584*cc^54 - 24233472560564*cc^53 - 38246082959208*cc^52 - 53525541512344*cc^51 - 68672842596586*cc^50 - 81046771293913*cc^49 - 93782956124025*cc^48 - 101664626355082*cc^47 - 106398474609511*cc^46 - 108209552862292*cc^45 - 114001665752135*cc^44 - 119457973759635*cc^43 - 117780728799265*cc^42 - 103134619148442*cc^41 - 87702419873019*cc^40 - 75889979373565*cc^39 - 61899407202381*cc^38 - 38577815134425*cc^37 - 14376503779568*cc^36 - 2040400169353*cc^35 + 2040400169353*cc^34 + 14376503779568*cc^33 + 38577815134425*cc^32 + 61899407202381*cc^31 + 75889979373565*cc^30 + 87702419873019*cc^29 + 103134619148442*cc^28 + 117780728799265*cc^27 + 119457973759635*cc^26 + 114001665752135*cc^25 + 108209552862292*cc^24 + 106398474609511*cc^23 + 101664626355082*cc^22 + 93782956124025*cc^21 + 81046771293913*cc^20 + 68672842596586*cc^19 + 53525541512344*cc^18 + 38246082959208*cc^17 + 24233472560564*cc^16 + 14600554466584*cc^15 + 7606453336976*cc^14 + 3681309755424*cc^13 + 1540921488048*cc^12 + 558887133344*cc^11 + 318179768512*cc^10 + 141495192896*cc^9 + 79472117632*cc^8 + 27145893376*cc^7 + 1962641664*cc^6 - 1189639680*cc^5 - 593355776*cc^4 - 312038400*cc^3 - 105560064*cc^2 - 20828160*cc - 1843200, 1843200*bb^69 + 20828160*bb^68 + 105560064*bb^67 + 312038400*bb^66 + 593355776*bb^65 + 1189639680*bb^64 - 1962641664*bb^63 - 27145893376*bb^62 - 79472117632*bb^61 - 141495192896*bb^60 - 318179768512*bb^59 - 558887133344*bb^58 - 1540921488048*bb^57 - 3681309755424*bb^56 - 7606453336976*bb^55 - 14600554466584*bb^54 - 24233472560564*bb^53 - 38246082959208*bb^52 - 53525541512344*bb^51 - 68672842596586*bb^50 - 81046771293913*bb^49 - 93782956124025*bb^48 - 101664626355082*bb^47 - 106398474609511*bb^46 - 108209552862292*bb^45 - 114001665752135*bb^44 - 119457973759635*bb^43 - 117780728799265*bb^42 - 103134619148442*bb^41 - 87702419873019*bb^40 - 75889979373565*bb^39 - 61899407202381*bb^38 - 38577815134425*bb^37 - 14376503779568*bb^36 - 2040400169353*bb^35 + 2040400169353*bb^34 + 14376503779568*bb^33 + 38577815134425*bb^32 + 61899407202381*bb^31 + 75889979373565*bb^30 + 87702419873019*bb^29 + 103134619148442*bb^28 + 117780728799265*bb^27 + 119457973759635*bb^26 + 114001665752135*bb^25 + 108209552862292*bb^24 + 106398474609511*bb^23 + 101664626355082*bb^22 + 93782956124025*bb^21 + 81046771293913*bb^20 + 68672842596586*bb^19 + 53525541512344*bb^18 + 38246082959208*bb^17 + 24233472560564*bb^16 + 14600554466584*bb^15 + 7606453336976*bb^14 + 3681309755424*bb^13 + 1540921488048*bb^12 + 558887133344*bb^11 + 318179768512*bb^10 + 141495192896*bb^9 + 79472117632*bb^8 + 27145893376*bb^7 + 1962641664*bb^6 - 1189639680*bb^5 - 593355776*bb^4 - 312038400*bb^3 - 105560064*bb^2 - 20828160*bb - 1843200, 1843200*c^69 + 20828160*c^68 + 105560064*c^67 + 312038400*c^66 + 593355776*c^65 + 1189639680*c^64 - 1962641664*c^63 - 27145893376*c^62 - 79472117632*c^61 - 141495192896*c^60 - 318179768512*c^59 - 558887133344*c^58 - 1540921488048*c^57 - 3681309755424*c^56 - 7606453336976*c^55 - 14600554466584*c^54 - 24233472560564*c^53 - 38246082959208*c^52 - 53525541512344*c^51 - 68672842596586*c^50 - 81046771293913*c^49 - 93782956124025*c^48 - 101664626355082*c^47 - 106398474609511*c^46 - 108209552862292*c^45 - 114001665752135*c^44 - 119457973759635*c^43 - 117780728799265*c^42 - 103134619148442*c^41 - 87702419873019*c^40 - 75889979373565*c^39 - 61899407202381*c^38 - 38577815134425*c^37 - 14376503779568*c^36 - 2040400169353*c^35 + 2040400169353*c^34 + 14376503779568*c^33 + 38577815134425*c^32 + 61899407202381*c^31 + 75889979373565*c^30 + 87702419873019*c^29 + 103134619148442*c^28 + 117780728799265*c^27 + 119457973759635*c^26 + 114001665752135*c^25 + 108209552862292*c^24 + 106398474609511*c^23 + 101664626355082*c^22 + 93782956124025*c^21 + 81046771293913*c^20 + 68672842596586*c^19 + 53525541512344*c^18 + 38246082959208*c^17 + 24233472560564*c^16 + 14600554466584*c^15 + 7606453336976*c^14 + 3681309755424*c^13 + 1540921488048*c^12 + 558887133344*c^11 + 318179768512*c^10 + 141495192896*c^9 + 79472117632*c^8 + 27145893376*c^7 + 1962641664*c^6 - 1189639680*c^5 - 593355776*c^4 - 312038400*c^3 - 105560064*c^2 - 20828160*c - 1843200, 1843200*b^69 + 20828160*b^68 + 105560064*b^67 + 312038400*b^66 + 593355776*b^65 + 1189639680*b^64 - 1962641664*b^63 - 27145893376*b^62 - 79472117632*b^61 - 141495192896*b^60 - 318179768512*b^59 - 558887133344*b^58 - 1540921488048*b^57 - 3681309755424*b^56 - 7606453336976*b^55 - 14600554466584*b^54 - 24233472560564*b^53 - 38246082959208*b^52 - 53525541512344*b^51 - 68672842596586*b^50 - 81046771293913*b^49 - 93782956124025*b^48 - 101664626355082*b^47 - 106398474609511*b^46 - 108209552862292*b^45 - 114001665752135*b^44 - 119457973759635*b^43 - 117780728799265*b^42 - 103134619148442*b^41 - 87702419873019*b^40 - 75889979373565*b^39 - 61899407202381*b^38 - 38577815134425*b^37 - 14376503779568*b^36 - 2040400169353*b^35 + 2040400169353*b^34 + 14376503779568*b^33 + 38577815134425*b^32 + 61899407202381*b^31 + 75889979373565*b^30 + 87702419873019*b^29 + 103134619148442*b^28 + 117780728799265*b^27 + 119457973759635*b^26 + 114001665752135*b^25 + 108209552862292*b^24 + 106398474609511*b^23 + 101664626355082*b^22 + 93782956124025*b^21 + 81046771293913*b^20 + 68672842596586*b^19 + 53525541512344*b^18 + 38246082959208*b^17 + 24233472560564*b^16 + 14600554466584*b^15 + 7606453336976*b^14 + 3681309755424*b^13 + 1540921488048*b^12 + 558887133344*b^11 + 318179768512*b^10 + 141495192896*b^9 + 79472117632*b^8 + 27145893376*b^7 + 1962641664*b^6 - 1189639680*b^5 - 593355776*b^4 - 312038400*b^3 - 105560064*b^2 - 20828160*b - 1843200) of Multivariate Polynomial Ring in b, c, d, e, f, bb, cc, dd, ee, ff over Rational Field On Wed, Aug 24, 2011 at 12:13 PM, Victor Miller <victorsmiller@gmail.com> wrote:
Indeed when I added equations like b*bb-1 == 0, to forbid b from being zero, the resulting ideal is 0 dimensional (which means that there are only a finite number of solutions over C). The groebner basis (in lex order) has 225 elements. Right now I'm attempting to calculate the radical and primary decomposition.
Victor
On Wed, Aug 24, 2011 at 12:09 PM, Victor Miller <victorsmiller@gmail.com> wrote:
When you set b=1 SAGE (actually Singular) almost immediately calculates the primary decomposition:
[Ideal (f, e, b - 1) of Multivariate Polynomial Ring in b, c, d, e, f over Rational Field, Ideal (f, c, b - 1) of Multivariate Polynomial Ring in b, c, d, e, f over Rational Field, Ideal (e, c, b - 1) of Multivariate Polynomial Ring in b, c, d, e, f over Rational Field, Ideal (f - 1, e - 1, d - 1, b - 1, c^2 + 5*c + 1) of Multivariate Polynomial Ring in b, c, d, e, f over Rational Field, Ideal (f - 1, d - 1, c - 1, b - 1, e^2 + 5*e + 1) of Multivariate Polynomial Ring in b, c, d, e, f over Rational Field, Ideal (e - 1, d - 1, c - 1, b - 1, f^2 + 5*f + 1) of Multivariate Polynomial Ring in b, c, d, e, f over Rational Field, Ideal (e, d, b - 1) of Multivariate Polynomial Ring in b, c, d, e, f over Rational Field, Ideal (f - 1, e - 1, c - 1, b - 1, d^2 + 5*d + 1) of Multivariate Polynomial Ring in b, c, d, e, f over Rational Field, Ideal (f, d, b - 1) of Multivariate Polynomial Ring in b, c, d, e, f over Rational Field, Ideal (d, c, b - 1) of Multivariate Polynomial Ring in b, c, d, e, f over Rational Field]
I didn't put in equations to forbid b,c,d,e,f from being 0. That seems to be the source of the higher dimensionality. A typical non-spurious component is b=1,d=1,e=1,f=1, c^2 + 5*c + 1 = 0.
On Wed, Aug 24, 2011 at 11:23 AM, Schroeppel, Richard <rschroe@sandia.gov> wrote:
Just add three more equations: Try b=c=d=1 first; if that's singular, try things like b=.6 or b+c=1.5.
Rich ________________________________________ From: math-fun-bounces@mailman.xmission.com [math-fun-bounces@mailman.xmission.com] on behalf of Bill Gosper [billgosper@gmail.com] Sent: Tuesday, August 23, 2011 11:56 PM To: math-fun@mailman.xmission.com Subject: Re: [math-fun] computer algebra
Victor Miller> Veit, I gave this to SAGE (which actually uses the Groeber Basis stuff
in SINGULAR) and it fairly quickly calculated a Groebner Basis, and showed that the dimension of the ideal is 3.
I.e., triply underdetermined?
That scotches the plan to PSLQ the exact algebraics from 1000 digit approximations. --rwg
Right now I'm waiting for it to produce a primary decomposition which should shed some light on the matter.
Victor
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Here's the factorization (over Q) of the groebner basis of the radical ideal. I've introduced extra variables I call bb,cc,dd,ee,ff which are the inverses of b,c,d,e,f respectively, enforced by equations like b*bb-1==0. The solutions are obtained by setting all of these polynomials to 0. Victor [f*ff - 1, e*ee - 1, d*dd - 1, c*cc - 1, b*bb - 1, b*c*d*e*f^2 + b^2*c*d*f + b*c^2*e*f + b*c*d*e*f + b*d^2*e*f + c*d*e^2*f + b*c*d*e, b*c*d*e^2*f + b^2*c*d*e + b*c*d^2*f + b*c*d*e*f + c^2*d*e*f + b*d*e*f^2 + b*c*e*f, b*c*d^2*e*f + b*c^2*d*e + b^2*d*e*f + b*c*d*e*f + b*c*e^2*f + c*d*e*f^2 + b*c*d*f, b*c^2*d*e*f + b*c*d^2*e + b^2*c*e*f + b*c*d*e*f + b*d*e^2*f + b*c*d*f^2 + c*d*e*f, b^2*c*d*e*f + b*c*d*e^2 + b*c^2*d*f + b*c*d*e*f + c*d^2*e*f + b*c*e*f^2 + b*d*e*f, (ff - 1) * (ff^2 - ff + 1) * (ff^2 + ff + 1) * (ff^2 + 5*ff + 1) * (5*ff^2 - 17*ff + 5) * (5*ff^2 + 11*ff + 5) * (ff^6 + ff^5 + ff^4 + ff^3 + ff^2 + ff + 1) * (4*ff^6 + 4*ff^5 + 32*ff^4 - 17*ff^3 + 32*ff^2 + 4*ff + 4) * (4*ff^18 + 12*ff^17 + 162*ff^16 + 321*ff^15 - 288*ff^14 - 1866*ff^13 - 4056*ff^12 - 6618*ff^11 - 8362*ff^10 - 8794*ff^9 - 8362*ff^8 - 6618*ff^7 - 4056*ff^6 - 1866*ff^5 - 288*ff^4 + 321*ff^3 + 162*ff^2 + 12*ff + 4), (f - 1) * (f^2 - f + 1) * (f^2 + f + 1) * (f^2 + 5*f + 1) * (5*f^2 - 17*f + 5) * (5*f^2 + 11*f + 5) * (f^6 + f^5 + f^4 + f^3 + f^2 + f + 1) * (4*f^6 + 4*f^5 + 32*f^4 - 17*f^3 + 32*f^2 + 4*f + 4) * (4*f^18 + 12*f^17 + 162*f^16 + 321*f^15 - 288*f^14 - 1866*f^13 - 4056*f^12 - 6618*f^11 - 8362*f^10 - 8794*f^9 - 8362*f^8 - 6618*f^7 - 4056*f^6 - 1866*f^5 - 288*f^4 + 321*f^3 + 162*f^2 + 12*f + 4), (ee - 1) * (ee + 1) * (ee^2 + ee + 1) * (ee^2 + 5*ee + 1) * (2*ee^2 + 3*ee + 2) * (ee^6 + ee^5 - 6*ee^4 - 13*ee^3 - 6*ee^2 + ee + 1) * (ee^6 + ee^5 + ee^4 + ee^3 + ee^2 + ee + 1) * (36*ee^8 - 36*ee^7 + 378*ee^6 + 294*ee^5 + 1057*ee^4 + 294*ee^3 + 378*ee^2 - 36*ee + 36) * (16*ee^36 + 96*ee^35 + 796*ee^34 + 1804*ee^33 + 17566*ee^32 + 38418*ee^31 + 165145*ee^30 + 451622*ee^29 + 1275250*ee^28 + 2800516*ee^27 + 6052848*ee^26 + 11530332*ee^25 + 19573592*ee^24 + 29211804*ee^23 + 39747080*ee^22 + 49794120*ee^21 + 58055092*ee^20 + 63387800*ee^19 + 65199950*ee^18 + 63387800*ee^17 + 58055092*ee^16 + 49794120*ee^15 + 39747080*ee^14 + 29211804*ee^13 + 19573592*ee^12 + 11530332*ee^11 + 6052848*ee^10 + 2800516*ee^9 + 1275250*ee^8 + 451622*ee^7 + 165145*ee^6 + 38418*ee^5 + 17566*ee^4 + 1804*ee^3 + 796*ee^2 + 96*ee + 16), (dd - 1) * (dd + 1) * (dd^2 + dd + 1) * (dd^2 + 5*dd + 1) * (2*dd^2 + 3*dd + 2) * (dd^6 + dd^5 - 6*dd^4 - 13*dd^3 - 6*dd^2 + dd + 1) * (dd^6 + dd^5 + dd^4 + dd^3 + dd^2 + dd + 1) * (36*dd^8 - 36*dd^7 + 378*dd^6 + 294*dd^5 + 1057*dd^4 + 294*dd^3 + 378*dd^2 - 36*dd + 36) * (16*dd^36 + 96*dd^35 + 796*dd^34 + 1804*dd^33 + 17566*dd^32 + 38418*dd^31 + 165145*dd^30 + 451622*dd^29 + 1275250*dd^28 + 2800516*dd^27 + 6052848*dd^26 + 11530332*dd^25 + 19573592*dd^24 + 29211804*dd^23 + 39747080*dd^22 + 49794120*dd^21 + 58055092*dd^20 + 63387800*dd^19 + 65199950*dd^18 + 63387800*dd^17 + 58055092*dd^16 + 49794120*dd^15 + 39747080*dd^14 + 29211804*dd^13 + 19573592*dd^12 + 11530332*dd^11 + 6052848*dd^10 + 2800516*dd^9 + 1275250*dd^8 + 451622*dd^7 + 165145*dd^6 + 38418*dd^5 + 17566*dd^4 + 1804*dd^3 + 796*dd^2 + 96*dd + 16), (e - 1) * (e + 1) * (e^2 + e + 1) * (e^2 + 5*e + 1) * (2*e^2 + 3*e + 2) * (e^6 + e^5 - 6*e^4 - 13*e^3 - 6*e^2 + e + 1) * (e^6 + e^5 + e^4 + e^3 + e^2 + e + 1) * (36*e^8 - 36*e^7 + 378*e^6 + 294*e^5 + 1057*e^4 + 294*e^3 + 378*e^2 - 36*e + 36) * (16*e^36 + 96*e^35 + 796*e^34 + 1804*e^33 + 17566*e^32 + 38418*e^31 + 165145*e^30 + 451622*e^29 + 1275250*e^28 + 2800516*e^27 + 6052848*e^26 + 11530332*e^25 + 19573592*e^24 + 29211804*e^23 + 39747080*e^22 + 49794120*e^21 + 58055092*e^20 + 63387800*e^19 + 65199950*e^18 + 63387800*e^17 + 58055092*e^16 + 49794120*e^15 + 39747080*e^14 + 29211804*e^13 + 19573592*e^12 + 11530332*e^11 + 6052848*e^10 + 2800516*e^9 + 1275250*e^8 + 451622*e^7 + 165145*e^6 + 38418*e^5 + 17566*e^4 + 1804*e^3 + 796*e^2 + 96*e + 16), (d - 1) * (d + 1) * (d^2 + d + 1) * (d^2 + 5*d + 1) * (2*d^2 + 3*d + 2) * (d^6 + d^5 - 6*d^4 - 13*d^3 - 6*d^2 + d + 1) * (d^6 + d^5 + d^4 + d^3 + d^2 + d + 1) * (36*d^8 - 36*d^7 + 378*d^6 + 294*d^5 + 1057*d^4 + 294*d^3 + 378*d^2 - 36*d + 36) * (16*d^36 + 96*d^35 + 796*d^34 + 1804*d^33 + 17566*d^32 + 38418*d^31 + 165145*d^30 + 451622*d^29 + 1275250*d^28 + 2800516*d^27 + 6052848*d^26 + 11530332*d^25 + 19573592*d^24 + 29211804*d^23 + 39747080*d^22 + 49794120*d^21 + 58055092*d^20 + 63387800*d^19 + 65199950*d^18 + 63387800*d^17 + 58055092*d^16 + 49794120*d^15 + 39747080*d^14 + 29211804*d^13 + 19573592*d^12 + 11530332*d^11 + 6052848*d^10 + 2800516*d^9 + 1275250*d^8 + 451622*d^7 + 165145*d^6 + 38418*d^5 + 17566*d^4 + 1804*d^3 + 796*d^2 + 96*d + 16), (cc - 1) * (cc^2 - cc + 1) * (cc^2 + cc + 1) * (cc^2 + 5*cc + 1) * (2*cc^2 + 3*cc + 2) * (5*cc^2 - 17*cc + 5) * (5*cc^2 + 11*cc + 5) * (cc^6 + cc^5 + cc^4 + cc^3 + cc^2 + cc + 1) * (4*cc^6 + 4*cc^5 + 18*cc^4 + 11*cc^3 + 18*cc^2 + 4*cc + 4) * (36*cc^8 - 36*cc^7 + 378*cc^6 + 294*cc^5 + 1057*cc^4 + 294*cc^3 + 378*cc^2 - 36*cc + 36) * (256*cc^36 + 1536*cc^35 + 3648*cc^34 - 7232*cc^33 - 21824*cc^32 + 76448*cc^31 + 121232*cc^30 - 418008*cc^29 + 454380*cc^28 + 457840*cc^27 - 1045658*cc^26 + 226204*cc^25 + 1449103*cc^24 - 1215674*cc^23 - 647357*cc^22 + 1855998*cc^21 - 514038*cc^20 - 796110*cc^19 + 1308641*cc^18 - 796110*cc^17 - 514038*cc^16 + 1855998*cc^15 - 647357*cc^14 - 1215674*cc^13 + 1449103*cc^12 + 226204*cc^11 - 1045658*cc^10 + 457840*cc^9 + 454380*cc^8 - 418008*cc^7 + 121232*cc^6 + 76448*cc^5 - 21824*cc^4 - 7232*cc^3 + 3648*cc^2 + 1536*cc + 256), (bb - 1) * (bb^2 - bb + 1) * (bb^2 + bb + 1) * (bb^2 + 5*bb + 1) * (2*bb^2 + 3*bb + 2) * (5*bb^2 - 17*bb + 5) * (5*bb^2 + 11*bb + 5) * (bb^6 + bb^5 + bb^4 + bb^3 + bb^2 + bb + 1) * (4*bb^6 + 4*bb^5 + 18*bb^4 + 11*bb^3 + 18*bb^2 + 4*bb + 4) * (36*bb^8 - 36*bb^7 + 378*bb^6 + 294*bb^5 + 1057*bb^4 + 294*bb^3 + 378*bb^2 - 36*bb + 36) * (256*bb^36 + 1536*bb^35 + 3648*bb^34 - 7232*bb^33 - 21824*bb^32 + 76448*bb^31 + 121232*bb^30 - 418008*bb^29 + 454380*bb^28 + 457840*bb^27 - 1045658*bb^26 + 226204*bb^25 + 1449103*bb^24 - 1215674*bb^23 - 647357*bb^22 + 1855998*bb^21 - 514038*bb^20 - 796110*bb^19 + 1308641*bb^18 - 796110*bb^17 - 514038*bb^16 + 1855998*bb^15 - 647357*bb^14 - 1215674*bb^13 + 1449103*bb^12 + 226204*bb^11 - 1045658*bb^10 + 457840*bb^9 + 454380*bb^8 - 418008*bb^7 + 121232*bb^6 + 76448*bb^5 - 21824*bb^4 - 7232*bb^3 + 3648*bb^2 + 1536*bb + 256), (c - 1) * (c^2 - c + 1) * (c^2 + c + 1) * (c^2 + 5*c + 1) * (2*c^2 + 3*c + 2) * (5*c^2 - 17*c + 5) * (5*c^2 + 11*c + 5) * (c^6 + c^5 + c^4 + c^3 + c^2 + c + 1) * (4*c^6 + 4*c^5 + 18*c^4 + 11*c^3 + 18*c^2 + 4*c + 4) * (36*c^8 - 36*c^7 + 378*c^6 + 294*c^5 + 1057*c^4 + 294*c^3 + 378*c^2 - 36*c + 36) * (256*c^36 + 1536*c^35 + 3648*c^34 - 7232*c^33 - 21824*c^32 + 76448*c^31 + 121232*c^30 - 418008*c^29 + 454380*c^28 + 457840*c^27 - 1045658*c^26 + 226204*c^25 + 1449103*c^24 - 1215674*c^23 - 647357*c^22 + 1855998*c^21 - 514038*c^20 - 796110*c^19 + 1308641*c^18 - 796110*c^17 - 514038*c^16 + 1855998*c^15 - 647357*c^14 - 1215674*c^13 + 1449103*c^12 + 226204*c^11 - 1045658*c^10 + 457840*c^9 + 454380*c^8 - 418008*c^7 + 121232*c^6 + 76448*c^5 - 21824*c^4 - 7232*c^3 + 3648*c^2 + 1536*c + 256), (b - 1) * (b^2 - b + 1) * (b^2 + b + 1) * (b^2 + 5*b + 1) * (2*b^2 + 3*b + 2) * (5*b^2 - 17*b + 5) * (5*b^2 + 11*b + 5) * (b^6 + b^5 + b^4 + b^3 + b^2 + b + 1) * (4*b^6 + 4*b^5 + 18*b^4 + 11*b^3 + 18*b^2 + 4*b + 4) * (36*b^8 - 36*b^7 + 378*b^6 + 294*b^5 + 1057*b^4 + 294*b^3 + 378*b^2 - 36*b + 36) * (256*b^36 + 1536*b^35 + 3648*b^34 - 7232*b^33 - 21824*b^32 + 76448*b^31 + 121232*b^30 - 418008*b^29 + 454380*b^28 + 457840*b^27 - 1045658*b^26 + 226204*b^25 + 1449103*b^24 - 1215674*b^23 - 647357*b^22 + 1855998*b^21 - 514038*b^20 - 796110*b^19 + 1308641*b^18 - 796110*b^17 - 514038*b^16 + 1855998*b^15 - 647357*b^14 - 1215674*b^13 + 1449103*b^12 + 226204*b^11 - 1045658*b^10 + 457840*b^9 + 454380*b^8 - 418008*b^7 + 121232*b^6 + 76448*b^5 - 21824*b^4 - 7232*b^3 + 3648*b^2 + 1536*b + 256)]
Thanks Victor! The complex Hadamard matrices of form {{1, 1, 1, 1, 1, 1, 1}, {1, -1 - b - c - d - e - f, b, c, d, e, f}, {1, b, b^2/d, b c, (b d)/f, (b e)/c, (b f)/e}, {1, c, b c, c^2/e, (c d)/b, (c e)/f, (c f)/d}, {1, d, (b d)/f, (c d)/b, d^2/e, d e, (d f)/c}, {1, e, (b e)/c, (c e)/f, d e, e^2/d, (e f)/b}, {1, f, (b f)/e, (c f)/d, (d f)/c, (e f)/b, f^2}} are now completely classified. I took your univariate polynomials in b (same as c), d (same as e), and f and found all the roots that have modulus 1. I then brute-force evaluated the polynomials {b c d e + b^2 c d f + b c^2 e f + b c d e f + b d^2 e f + c d e^2 f + b c d e f^2, b^2 c d e + b c d^2 f + b c e f + b c d e f + c^2 d e f + b c d e^2 f + b d e f^2, b c^2 d e + b c d f + b^2 d e f + b c d e f + b c d^2 e f + b c e^2 f + c d e f^2, b c d^2 e + b^2 c e f + c d e f + b c d e f + b c^2 d e f + b d e^2 f + b c d f^2, b c d e^2 + b c^2 d f + b d e f + b c d e f + b^2 c d e f + c d^2 e f + b c e f^2} at all combinations of the modulus 1 roots and identified those combinations on which all five of the above vanished. This gave 48 solutions overall. There were no surprises, but at least now I have algebraic expressions for the exotic ones (roots of polynomials of degree 18 and 36). There is a website run by polish academics that records these things and I will credit you for this contribution. Veit On Aug 24, 2011, at 1:00 PM, Victor Miller wrote:
Here's the factorization (over Q) of the groebner basis of the radical ideal. I've introduced extra variables I call bb,cc,dd,ee,ff which are the inverses of b,c,d,e,f respectively, enforced by equations like b*bb-1==0. The solutions are obtained by setting all of these polynomials to 0.
Victor
Veit, You're welcome. It's an interesting problem. What's the URL of the website? Victor On Wed, Aug 24, 2011 at 5:13 PM, Veit Elser <ve10@cornell.edu> wrote:
Thanks Victor! The complex Hadamard matrices of form
{{1, 1, 1, 1, 1, 1, 1}, {1, -1 - b - c - d - e - f, b, c, d, e, f}, {1, b, b^2/d, b c, (b d)/f, (b e)/c, (b f)/e}, {1, c, b c, c^2/e, (c d)/b, (c e)/f, (c f)/d}, {1, d, (b d)/f, (c d)/b, d^2/e, d e, (d f)/c}, {1, e, (b e)/c, (c e)/f, d e, e^2/d, (e f)/b}, {1, f, (b f)/e, (c f)/d, (d f)/c, (e f)/b, f^2}}
are now completely classified. I took your univariate polynomials in b (same as c), d (same as e), and f and found all the roots that have modulus 1. I then brute-force evaluated the polynomials
{b c d e + b^2 c d f + b c^2 e f + b c d e f + b d^2 e f + c d e^2 f + b c d e f^2, b^2 c d e + b c d^2 f + b c e f + b c d e f + c^2 d e f + b c d e^2 f + b d e f^2, b c^2 d e + b c d f + b^2 d e f + b c d e f + b c d^2 e f + b c e^2 f + c d e f^2, b c d^2 e + b^2 c e f + c d e f + b c d e f + b c^2 d e f + b d e^2 f + b c d f^2, b c d e^2 + b c^2 d f + b d e f + b c d e f + b^2 c d e f + c d^2 e f + b c e f^2}
at all combinations of the modulus 1 roots and identified those combinations on which all five of the above vanished. This gave 48 solutions overall. There were no surprises, but at least now I have algebraic expressions for the exotic ones (roots of polynomials of degree 18 and 36).
There is a website run by polish academics that records these things and I will credit you for this contribution.
Veit
On Aug 24, 2011, at 1:00 PM, Victor Miller wrote:
Here's the factorization (over Q) of the groebner basis of the radical ideal. I've introduced extra variables I call bb,cc,dd,ee,ff which are the inverses of b,c,d,e,f respectively, enforced by equations like b*bb-1==0. The solutions are obtained by setting all of these polynomials to 0.
Victor
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On Aug 24, 2011, at 5:42 PM, Victor Miller wrote:
Veit, You're welcome. It's an interesting problem. What's the URL of the website?
Victor
Veit, Thanks. I didn't know anything about complex Hadamard matrices. One thing that I find quite interesting is the relationship between them and tiling of finite abelian groups. I have a recent paper about such a problem. Now I'm wondering about the relationship between that problem (whether or no a subset tiles the group Z_2^n) and the existence of complex Hadamard matrices. Victor On Wed, Aug 24, 2011 at 8:45 PM, Veit Elser <ve10@cornell.edu> wrote:
On Aug 24, 2011, at 5:42 PM, Victor Miller wrote:
Veit, You're welcome. It's an interesting problem. What's the URL of the website?
Victor
http://chaos.if.uj.edu.pl/~karol/hadamard/
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Victor, It looks like there is a connection to your problem, see: http://chaos.if.uj.edu.pl/~karol/hadamard/chm_papers/MRS06_constructions.pdf I didn't find your paper on arxiv. Veit On Aug 24, 2011, at 10:21 PM, Victor Miller wrote:
Veit, Thanks. I didn't know anything about complex Hadamard matrices. One thing that I find quite interesting is the relationship between them and tiling of finite abelian groups. I have a recent paper about such a problem. Now I'm wondering about the relationship between that problem (whether or no a subset tiles the group Z_2^n) and the existence of complex Hadamard matrices.
Victor
participants (4)
-
Bill Gosper -
Schroeppel, Richard -
Veit Elser -
Victor Miller