Tropical-To-Toric Embedding Report

This page is a lightweight native-rendered mirror of the selected exact analysis artifact. Sage, Macaulay2, antichain selection, toric-ideal relations, and Miller-Sturmfels resolution data are computed before page generation; the browser only draws precomputed JSON and handles simple hover/click interactions.

manifest JSON

No demo sidecars are accepted. The displayed staircase is derived from the checkpoint embedding payload through exact product-order antichain selection, then certified by Sage and Macaulay2.

Interactive Miller-Sturmfels Staircase

This native viewer draws the exact precomputed quotient-basis lattice points, minimal generators, adjacent LCM layer, and ideal region. The overhead mode shows the Miller-Sturmfels xy-grid; the projected 3D mode lifts module layers by small heights while keeping the same underlying exact lattice data.

Click a lattice point, generator, or LCM corner for exact monomial data.

record_000

exponents7
dimension2
Sage normal fan exactTrue
Macaulay2 toric ideal exactTrue
selectionstaircase_antichain
input unique checkpoint exponents435
minimal generators[[25, 0], [24, 6], [17, 27], [16, 30], [13, 32], [2, 34], [0, 48]]
quotient-basis lattice points791
adjacent LCM corners6
Macaulay2 toric ideal generators96
Macaulay2 resolution length7
Macaulay2 regularity35

summary JSONexact CAS sidecar JSON

Complete Macaulay2 toric ideal relation list (96)
  1. x_1^2*x_5^2*x_6-x_4^4
  2. x_0*x_1*x_4*x_5*x_6-x_3^4
  3. x_2^2*x_5^4-x_3*x_4^2*x_6^2
  4. x_2^2*x_4^2*x_5^2-x_1^2*x_3*x_6^3
  5. x_0^3*x_3*x_5^2-x_1^2*x_2^2*x_4
  6. x_0*x_4^5-x_1*x_3^4*x_5
  7. x_0^3*x_3*x_4^3-x_1^4*x_2^2*x_6
  8. x_0*x_1^3*x_6^4-x_2^2*x_3^3*x_4*x_5
  9. x_0^3*x_3^2*x_6^3-x_2^4*x_4^3
  10. x_0^4*x_5^3*x_6-x_1*x_2^2*x_3^3
  11. x_0^3*x_4*x_5^4-x_1^4*x_6^3
  12. x_2^2*x_3^3*x_5^3-x_0*x_1*x_4^3*x_6^3
  13. x_1^4*x_3*x_5^3-x_0^2*x_2^4*x_6
  14. x_1^2*x_2^4*x_5^2-x_0^3*x_3^2*x_4*x_6^2
  15. x_1^2*x_3*x_4^4*x_5-x_0^2*x_2^4*x_6^2
  16. x_1^6*x_4*x_5-x_0^5*x_2^2*x_6
  17. x_0^2*x_3^5*x_5-x_1^3*x_2^2*x_4^2*x_6
  18. x_2^8-x_1*x_3^7
  19. x_0^4*x_4^2*x_6^3-x_1*x_2^4*x_3^2*x_5
  20. x_0^3*x_5^6-x_1^2*x_4^3*x_6^2
  21. x_1^2*x_4^2*x_5^5-x_0^2*x_2^2*x_6^4
  22. x_4^6*x_5^3-x_0^2*x_2^2*x_6^5
  23. x_0*x_3^2*x_4^3*x_5^3-x_2^6*x_6^2
  24. x_0^2*x_3^4*x_5^3-x_1^5*x_6^4
  25. x_0^4*x_4^4*x_5-x_1^3*x_2^2*x_3^3
  26. x_0^2*x_2^6*x_5-x_1^4*x_3^2*x_4^2*x_6
  27. x_0^4*x_1*x_5*x_6^4-x_2^4*x_3^2*x_4^2
  28. x_1^5*x_4^3*x_6^2-x_0*x_2^6*x_3^2
  29. x_0*x_3*x_4*x_5^7-x_2^4*x_6^4
  30. x_0*x_1^2*x_3^2*x_5^5-x_2^6*x_4*x_6
  31. x_1^5*x_4*x_5^4-x_0*x_2^4*x_3^3
  32. x_1^4*x_4^3*x_5^3-x_0^5*x_3*x_6^4
  33. x_1^3*x_4^5*x_5^2-x_0*x_2^4*x_3^3*x_6
  34. x_3^6*x_4^2*x_5^2-x_1*x_2^6*x_6^3
  35. x_1*x_2^4*x_4^4*x_5-x_0^2*x_3^6*x_6^2
  36. x_1*x_4^9-x_0*x_2^4*x_3^3*x_6^2
  37. x_0^4*x_1*x_6^6-x_2^6*x_3*x_5^3
  38. x_0^6*x_6^5-x_1^3*x_3^3*x_4^2*x_5^2
  39. x_1^6*x_3^2*x_5*x_6^2-x_0^2*x_2^6*x_4^2
  40. x_0^6*x_3^3*x_6^2-x_1^4*x_2^6
  41. x_0^8*x_3*x_5*x_6-x_1^8*x_4^2
  42. x_1^2*x_5^9-x_0^2*x_3*x_6^6
  43. x_4^4*x_5^7-x_0^2*x_3*x_6^7
  44. x_1^4*x_5^7-x_0^2*x_2^2*x_4^2*x_6^3
  45. x_3^5*x_5^6-x_1*x_2^4*x_6^5
  46. x_0^2*x_1^3*x_5^6-x_2^6*x_3^2
  47. x_1^6*x_5^5-x_0^5*x_3*x_4*x_6^3
  48. x_0^2*x_1*x_4^4*x_5^4-x_2^6*x_3^2*x_6
  49. x_1^8*x_5^3-x_0^5*x_2^2*x_4^3
  50. x_0*x_2^6*x_3^2*x_5^2-x_1^3*x_4^7*x_6
  51. x_0^5*x_2^4*x_4*x_5-x_1^8*x_3*x_6^2
  52. x_1^5*x_4^6-x_0^4*x_2^2*x_3^4*x_6
  53. x_0^8*x_3*x_4^2-x_1^10*x_5
  54. x_0^7*x_4*x_5*x_6^3-x_1^3*x_2^6*x_3
  55. x_1^9*x_6^3-x_0^4*x_2^4*x_3^3
  56. x_0*x_5^11-x_2^2*x_4*x_6^6
  57. x_1*x_3^3*x_5^8-x_0^3*x_4*x_6^7
  58. x_3^3*x_4^3*x_5^6-x_0^3*x_1*x_6^8
  59. x_0^5*x_2^4*x_5^3-x_1^6*x_3*x_4^3*x_6
  60. x_0*x_3^9*x_5^2-x_1^2*x_2^2*x_4^7*x_6
  61. x_0^3*x_2^6*x_4^3-x_1^5*x_3^6*x_6
  62. x_0^7*x_3^5-x_1^9*x_4^3
  63. x_0^9*x_5^2*x_6^2-x_1^7*x_3^3*x_4
  64. x_0^9*x_4^3*x_6-x_1^9*x_3^3
  65. x_0*x_4^3*x_5^9-x_1^2*x_2^2*x_6^7
  66. x_0^2*x_1*x_4^2*x_5^8-x_2^4*x_3^3*x_6^3
  67. x_3^5*x_4^4*x_5^4-x_1^3*x_2^4*x_6^6
  68. x_0*x_3^8*x_5^4-x_1^4*x_4^5*x_6^4
  69. x_0^7*x_4^3*x_5^3-x_1^5*x_2^4*x_3^2
  70. x_3^10*x_5^3-x_0*x_2^6*x_4^3*x_6^3
  71. x_0^8*x_2^2*x_5^3-x_1^10*x_6^2
  72. x_4^11*x_5^2-x_0*x_1*x_2^2*x_3^4*x_6^5
  73. x_1^5*x_2^6*x_4*x_5-x_0^5*x_3^7*x_6
  74. x_2^4*x_4^9-x_0*x_3^10*x_6^2
  75. x_0^9*x_4*x_6^4-x_1^7*x_2^2*x_3^2*x_5^2
  76. x_3^4*x_5^10-x_1*x_2^2*x_4^2*x_6^7
  77. x_3^4*x_4^2*x_5^8-x_1^3*x_2^2*x_6^8
  78. x_3^9*x_5^5-x_0*x_1^2*x_2^4*x_4*x_6^6
  79. x_4^14-x_2^4*x_3^7*x_5*x_6^2
  80. x_3^3*x_5^12-x_1^3*x_6^10
  81. x_0^11*x_3^2*x_4*x_5-x_1^12*x_2^2
  82. x_0^12*x_6^4-x_1^9*x_2^4*x_3
  83. x_1^4*x_2^6*x_4^6-x_0^4*x_3^11*x_6
  84. x_0^10*x_2^6-x_1^14*x_3*x_6
  85. x_0^12*x_4^2*x_5^2*x_6-x_1^11*x_2^2*x_3^2
  86. x_0^16*x_3^2*x_6-x_1^18
  87. x_0^13*x_2^4*x_5^2-x_1^16*x_4*x_6
  88. x_0^13*x_2^4*x_4^3-x_1^18*x_6^2
  89. x_0^17*x_4*x_5*x_6^2-x_1^17*x_3^2
  90. x_4^3*x_5^18-x_0*x_2^2*x_3*x_6^13
  91. x_1*x_3^2*x_4^3*x_5^15-x_0^5*x_6^14
  92. x_0^12*x_2^4*x_3^4*x_5-x_1^17*x_4^2*x_6^2
  93. x_4*x_5^22-x_0*x_3^2*x_6^15
  94. x_3^2*x_5^21-x_0^2*x_1*x_6^16
  95. x_1*x_3*x_5^30-x_0^4*x_6^22
  96. x_5^33-x_2^2*x_3^2*x_6^21

Miller-Sturmfels S/I Resolution

basisLCM shiftdifferentiald1 d2 check
f_0x^25 y^6d2(f_0) = y^6 e_0 - x e_1d1 d2(f_0) = x^25 y^6 - x^25 y^6 = 0
f_1x^24 y^27d2(f_1) = y^21 e_1 - x^7 e_2d1 d2(f_1) = x^24 y^27 - x^24 y^27 = 0
f_2x^17 y^30d2(f_2) = y^3 e_2 - x e_3d1 d2(f_2) = x^17 y^30 - x^17 y^30 = 0
f_3x^16 y^32d2(f_3) = y^2 e_3 - x^3 e_4d1 d2(f_3) = x^16 y^32 - x^16 y^32 = 0
f_4x^13 y^34d2(f_4) = y^2 e_4 - x^11 e_5d1 d2(f_4) = x^13 y^34 - x^13 y^34 = 0
f_5x^2 y^48d2(f_5) = y^14 e_5 - x^2 e_6d1 d2(f_5) = x^2 y^48 - x^2 y^48 = 0