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Hilberts Nullstellensatz, Linear Algebra and Combinatorial Problems Susan Margulies Mathematics Department, US Naval Academy, Annapolis, Maryland Visiting National Institute of Standards and Technology! April 7, 2015 Susan Margulies, US

  1. Hilbert’s Nullstellensatz Theorem (1893): Let K be an algebraically closed field and f 1 , . . . , f s be polynomials in K [ x 1 , . . . , x n ]. Given a system of equations such that f 1 = f 2 = · · · = f s = 0 , then this system has no solution if and only if there exist polynomials β 1 , . . . , β s ∈ K [ x 1 , . . . , x n ] such that s � 1 = β i f i . ✷ i =1 � �� � This polynomial identity is a Nullstellensatz certificate . � � Definition: Let d = max deg( β 1 ) , deg( β 2 ) , . . . , deg( β s ) . Then d is the degree of the Nullstellensatz certificate . Susan Margulies, US Naval Academy NulLA and Combinatorial Infeasibility

  2. Nullstellensatz Degree Upper Bounds Recall n is the number of variables, and the number of monomials � n + d − 1 � of degree d in n variables is . n − 1 Theorem: (Koll´ ar, 1988) The deg( β i ) is bounded by � �� n � deg( β i ) ≤ max 3 , max { deg( f i ) } . (bound is tight for certain pathologically bad examples) Theorem: (Lazard 1977, Brownawell 1987) The deg( β i ) is bounded by � � deg( β i ) ≤ n max { deg( f i ) } − 1 . (bound applies to particular zero-dimensional ideals) Susan Margulies, US Naval Academy NulLA and Combinatorial Infeasibility

  3. Nullstellensatz Degree Upper Bounds Recall n is the number of variables, and the number of monomials � n + d − 1 � of degree d in n variables is . n − 1 Theorem: (Koll´ ar, 1988) The deg( β i ) is bounded by � �� n � deg( β i ) ≤ max 3 , max { deg( f i ) } . (bound is tight for certain pathologically bad examples) Theorem: (Lazard 1977, Brownawell 1987) The deg( β i ) is bounded by � � deg( β i ) ≤ n max { deg( f i ) } − 1 . (bound applies to particular zero-dimensional ideals) Question: What about lower bounds? How do we find them? Susan Margulies, US Naval Academy NulLA and Combinatorial Infeasibility

  4. N ul LA running on a particular instance: INPUT: A system of polynomial equations x 2 1 − 1 = 0 , x 1 + x 3 = 0 , x 1 + x 2 = 0 , x 2 + x 3 = 0 Susan Margulies, US Naval Academy NulLA and Combinatorial Infeasibility

  5. N ul LA running on a particular instance: INPUT: A system of polynomial equations x 2 1 − 1 = 0 , x 1 + x 3 = 0 , x 1 + x 2 = 0 , x 2 + x 3 = 0 1 Construct a hypothetical Nullstellensatz certificate of degree 1 ( x 2 1 = ( c 0 x 1 + c 1 x 2 + c 2 x 3 + c 3 ) 1 − 1) + ( c 4 x 1 + c 5 x 2 + c 6 x 3 + c 7 ) ( x 1 + x 2 ) � �� � � �� � β 1 β 2 + ( c 8 x 1 + c 9 x 2 + c 10 x 3 + c 11 ) ( x 1 + x 3 ) + ( c 12 x 1 + c 13 x 2 + c 14 x 3 + c 15 ) ( x 2 + x 3 ) � �� � � �� � β 3 β 4 Susan Margulies, US Naval Academy NulLA and Combinatorial Infeasibility

  6. N ul LA running on a particular instance: INPUT: A system of polynomial equations x 2 1 − 1 = 0 , x 1 + x 3 = 0 , x 1 + x 2 = 0 , x 2 + x 3 = 0 1 Construct a hypothetical Nullstellensatz certificate of degree 1 ( x 2 1 = ( c 0 x 1 + c 1 x 2 + c 2 x 3 + c 3 ) 1 − 1) + ( c 4 x 1 + c 5 x 2 + c 6 x 3 + c 7 ) ( x 1 + x 2 ) � �� � � �� � β 1 β 2 + ( c 8 x 1 + c 9 x 2 + c 10 x 3 + c 11 ) ( x 1 + x 3 ) + ( c 12 x 1 + c 13 x 2 + c 14 x 3 + c 15 ) ( x 2 + x 3 ) � �� � � �� � β 3 β 4 2 Expand the hypothetical Nullstellensatz certificate c 0 x 3 1 + c 1 x 2 1 x 2 + c 2 x 2 1 x 3 + ( c 3 + c 4 + c 8 ) x 2 1 + ( c 5 + c 13 ) x 2 2 + ( c 10 + c 14 ) x 2 3 + ( c 4 + c 5 + c 9 + c 12 ) x 1 x 2 + ( c 6 + c 8 + c 10 + c 12 ) x 1 x 3 + ( c 6 + c 9 + c 13 + c 14 ) x 2 x 3 + ( c 7 + c 11 − c 0 ) x 1 + ( c 7 + c 15 − c 1 ) x 2 + ( c 11 + c 15 − c 2 ) x 3 − c 3 Susan Margulies, US Naval Academy NulLA and Combinatorial Infeasibility

  7. N ul LA running on a particular instance: INPUT: A system of polynomial equations x 2 1 − 1 = 0 , x 1 + x 3 = 0 , x 1 + x 2 = 0 , x 2 + x 3 = 0 1 Construct a hypothetical Nullstellensatz certificate of degree 1 ( x 2 1 = ( c 0 x 1 + c 1 x 2 + c 2 x 3 + c 3 ) 1 − 1) + ( c 4 x 1 + c 5 x 2 + c 6 x 3 + c 7 ) ( x 1 + x 2 ) � �� � � �� � β 1 β 2 + ( c 8 x 1 + c 9 x 2 + c 10 x 3 + c 11 ) ( x 1 + x 3 ) + ( c 12 x 1 + c 13 x 2 + c 14 x 3 + c 15 ) ( x 2 + x 3 ) � �� � � �� � β 3 β 4 2 Expand the hypothetical Nullstellensatz certificate c 0 x 3 1 + c 1 x 2 1 x 2 + c 2 x 2 1 x 3 + ( c 3 + c 4 + c 8 ) x 2 1 + ( c 5 + c 13 ) x 2 2 + ( c 10 + c 14 ) x 2 3 + ( c 4 + c 5 + c 9 + c 12 ) x 1 x 2 + ( c 6 + c 8 + c 10 + c 12 ) x 1 x 3 + ( c 6 + c 9 + c 13 + c 14 ) x 2 x 3 + ( c 7 + c 11 − c 0 ) x 1 + ( c 7 + c 15 − c 1 ) x 2 + ( c 11 + c 15 − c 2 ) x 3 − c 3 3 Extract a linear system of equations from expanded certificate c 0 = 0 , . . . , c 3 + c 4 + c 8 = 0 , c 11 + c 15 − c 2 = 0 , − c 3 = 1 Susan Margulies, US Naval Academy NulLA and Combinatorial Infeasibility

  8. N ul LA running on a particular instance: c 0 c 1 c 2 c 3 c 4 c 5 c 6 c 7 c 8 c 9 c 10 c 11 c 12 c 13 c 14 c 15 x 3 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 x 2 1 x 2 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 x 2 1 x 3 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 x 2 0 0 0 1 1 0 0 0 1 0 0 0 0 0 0 0 0 1 x 2 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 2 x 2 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 3 0 0 0 0 1 1 0 0 0 1 0 0 1 0 0 0 0 x 1 x 2 0 0 0 0 0 0 1 0 1 0 1 0 1 0 0 0 0 x 1 x 3 0 0 0 0 0 0 1 0 0 1 0 0 0 1 1 0 0 x 2 x 3 − 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 x 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 x 2 0 0 − 1 0 0 0 0 0 0 0 0 1 0 0 0 1 0 x 3 1 0 0 0 − 1 0 0 0 0 0 0 0 0 0 0 0 0 1 Susan Margulies, US Naval Academy NulLA and Combinatorial Infeasibility

  9. N ul LA running on a particular instance: 4 Solve the linear system, and assemble the certificate 1 − 1) + 1 2 x 1 ( x 1 + x 2 ) − 1 2 x 1 ( x 2 + x 3 ) + 1 1 = − ( x 2 2 x 1 ( x 1 + x 3 ) Susan Margulies, US Naval Academy NulLA and Combinatorial Infeasibility

  10. N ul LA running on a particular instance: 4 Solve the linear system, and assemble the certificate 1 − 1) + 1 2 x 1 ( x 1 + x 2 ) − 1 2 x 1 ( x 2 + x 3 ) + 1 1 = − ( x 2 2 x 1 ( x 1 + x 3 ) 5 Otherwise, increment the degree and repeat. Susan Margulies, US Naval Academy NulLA and Combinatorial Infeasibility

  11. N ul LA Summary INPUT: A system of polynomial equations Construct a hypothetical Nullstellensatz certificate of degree d 1 Expand the hypothetical Nullstellensatz certificate 2 Extract a linear system of equations from expanded certificate 3 Solve the linear system. 4 If there is a solution, assemble the certificate. 1 Otherwise, loop and repeat with a larger degree d until known 2 upper bounds are exceeded. OUTPUT: yes , there is a solution. 1 no , there is no solution, along with a certificate of infeasibility . 2 Susan Margulies, US Naval Academy NulLA and Combinatorial Infeasibility

  12. Partition Problem: Definition and Example Partition: Given set of integers W = { w 1 , . . . , w n } , can W be partitioned into two sets, S and W \ S such that � � w = w . w ∈ S w ∈ W \ S Susan Margulies, US Naval Academy NulLA and Combinatorial Infeasibility

  13. Partition Problem: Definition and Example Partition: Given set of integers W = { w 1 , . . . , w n } , can W be partitioned into two sets, S and W \ S such that � � w = w . w ∈ S w ∈ W \ S Example: Let W = { 1 , 3 , 5 , 7 , 7 , 9 } . Then � �� � ���� S W \ S . Susan Margulies, US Naval Academy NulLA and Combinatorial Infeasibility

  14. Partition Problem: Definition and Example Partition: Given set of integers W = { w 1 , . . . , w n } , can W be partitioned into two sets, S and W \ S such that � � w = w . w ∈ S w ∈ W \ S Example: Let W = { 1 , 3 , 5 , 7 , 7 , 9 } . Then � �� � ���� S W \ S 1 + 3 + 5 + 7 . � �� � S Susan Margulies, US Naval Academy NulLA and Combinatorial Infeasibility

  15. Partition Problem: Definition and Example Partition: Given set of integers W = { w 1 , . . . , w n } , can W be partitioned into two sets, S and W \ S such that � � w = w . w ∈ S w ∈ W \ S Example: Let W = { 1 , 3 , 5 , 7 , 7 , 9 } . Then � �� � ���� S W \ S 1 + 3 + 5 + 7 7 + 9 . � �� � � �� � S W \ S Susan Margulies, US Naval Academy NulLA and Combinatorial Infeasibility

  16. Partition Problem: Definition and Example Partition: Given set of integers W = { w 1 , . . . , w n } , can W be partitioned into two sets, S and W \ S such that � � w = w . w ∈ S w ∈ W \ S Example: Let W = { 1 , 3 , 5 , 7 , 7 , 9 } . Then � �� � ���� S W \ S 16 = 1 + 3 + 5 + 7 = 7 + 9 = 16 . � �� � � �� � S W \ S Susan Margulies, US Naval Academy NulLA and Combinatorial Infeasibility

  17. Partition as a System of Polynomial Equations Given a set of integers W = { w 1 , . . . , w n } : one variable per integer : x 1 , . . . , x n Susan Margulies, US Naval Academy NulLA and Combinatorial Infeasibility

  18. Partition as a System of Polynomial Equations Given a set of integers W = { w 1 , . . . , w n } : one variable per integer : x 1 , . . . , x n For i = 1 , . . . , n , let x 2 i − 1 = 0 . Susan Margulies, US Naval Academy NulLA and Combinatorial Infeasibility

  19. Partition as a System of Polynomial Equations Given a set of integers W = { w 1 , . . . , w n } : one variable per integer : x 1 , . . . , x n For i = 1 , . . . , n , let x 2 i − 1 = 0 . and finally, n � w i x i = 0 . i =1 Susan Margulies, US Naval Academy NulLA and Combinatorial Infeasibility

  20. Partition as a System of Polynomial Equations Given a set of integers W = { w 1 , . . . , w n } : one variable per integer : x 1 , . . . , x n For i = 1 , . . . , n , let x 2 i − 1 = 0 . and finally, n � w i x i = 0 . i =1 Proposition: Given a set of integers W = { w 1 , . . . , w n } , the above system of n + 1 polynomial equations has a solution if and only if there exists a partition of W into two sets, S ⊆ W and W \ S , such that � w ∈ S w = � w ∈ W \ S w . Susan Margulies, US Naval Academy NulLA and Combinatorial Infeasibility

  21. Partition as a System of Polynomial Equations Given a set of integers W = { w 1 , . . . , w n } : one variable per integer : x 1 , . . . , x n For i = 1 , . . . , n , let x 2 i − 1 = 0 . and finally, n � w i x i = 0 . i =1 Proposition: Given a set of integers W = { w 1 , . . . , w n } , the above system of n + 1 polynomial equations has a solution if and only if there exists a partition of W into two sets, S ⊆ W and W \ S , such that � w ∈ S w = � w ∈ W \ S w . Question: Let W = { 1 , 3 , 5 , 2 } . Is W partitionable? x 2 x 2 x 3 x 2 1 − 1 = 0 , 2 − 1 = 0 , 3 − 1 = 0 , 4 − 1 = 0 , x 1 + 3 x 2 + 5 x 3 + 2 x 4 = 0 . Susan Margulies, US Naval Academy NulLA and Combinatorial Infeasibility

  22. NP, coNP and the Nullstellensatz Susan Margulies, US Naval Academy NulLA and Combinatorial Infeasibility

  23. NP, coNP and the Nullstellensatz Definition NP is the class of problems whose solutions can be verified in polynomial-time. Susan Margulies, US Naval Academy NulLA and Combinatorial Infeasibility

  24. NP, coNP and the Nullstellensatz Definition NP is the class of problems whose solutions can be verified in polynomial-time. Definition coNP is the class of problems whose complements are in NP . Susan Margulies, US Naval Academy NulLA and Combinatorial Infeasibility

  25. NP, coNP and the Nullstellensatz Definition NP is the class of problems whose solutions can be verified in polynomial-time. Definition coNP is the class of problems whose complements are in NP . It is widely believed that coNP � = NP . Susan Margulies, US Naval Academy NulLA and Combinatorial Infeasibility

  26. NP, coNP and the Nullstellensatz Definition NP is the class of problems whose solutions can be verified in polynomial-time. (hard to find) Definition coNP is the class of problems whose complements are in NP . It is widely believed that coNP � = NP . Susan Margulies, US Naval Academy NulLA and Combinatorial Infeasibility

  27. NP, coNP and the Nullstellensatz Definition NP is the class of problems whose solutions can be verified in polynomial-time. (hard to find) Definition coNP is the class of problems whose complements are in NP . (hard to verify) It is widely believed that coNP � = NP . Susan Margulies, US Naval Academy NulLA and Combinatorial Infeasibility

  28. NP, coNP and the Nullstellensatz Definition NP is the class of problems whose solutions can be verified in polynomial-time. (hard to find) Definition coNP is the class of problems whose complements are in NP . (hard to verify) Observation The Partition problem is NP-complete. It is widely believed that coNP � = NP . Susan Margulies, US Naval Academy NulLA and Combinatorial Infeasibility

  29. Minimum-degree Nullstellensatz Certificates Example Question: Let W = { 1 , 3 , 5 , 2 } . Is W partitionable? x 2 x 2 x 3 x 2 1 − 1 = 0 , 2 − 1 = 0 , 3 − 1 = 0 , 4 − 1 = 0 , x 1 + 3 x 2 + 5 x 3 + 2 x 4 = 0 . Susan Margulies, US Naval Academy NulLA and Combinatorial Infeasibility

  30. Minimum-degree Nullstellensatz Certificates Example Question: Let W = { 1 , 3 , 5 , 2 } . Is W partitionable? Answer: No! x 2 x 2 x 3 x 2 1 − 1 = 0 , 2 − 1 = 0 , 3 − 1 = 0 , 4 − 1 = 0 , x 1 + 3 x 2 + 5 x 3 + 2 x 4 = 0 . � � − 155 693 + 842 3465 x 2 x 3 − 188 693 x 2 x 4 + 908 ( x 2 1 = 1 − 1 ) 3465 x 3 x 4 � � 231 + 842 1 1155 x 1 x 3 − 188 231 x 1 x 4 + 292 ( x 2 + − 1155 x 3 x 4 2 − 1 ) � � − 467 693 + 842 693 x 1 x 2 + 908 693 x 1 x 4 + 292 ( x 2 + 693 x 2 x 4 3 − 1 ) � � − 68 693 − 376 693 x 1 x 2 + 1816 3465 x 1 x 3 + 584 ( x 2 + 3465 x 2 x 3 4 − 1 ) � 155 693 x 2 + 467 1 3465 x 3 + 34 693 x 4 − 842 + 693 x 1 + 3465 x 1 x 2 x 3 � + 188 693 x 1 x 2 x 4 − 908 3465 x 1 x 3 x 4 − 292 3465 x 2 x 3 x 4 ( x 1 + 3x 2 + 5x 3 + 2x 4 ) . Susan Margulies, US Naval Academy NulLA and Combinatorial Infeasibility

  31. Minimum-degree Partition Nullstellensatz Certificates Let S n k denote the set of k -subsets of { 1 , . . . , n } Susan Margulies, US Naval Academy NulLA and Combinatorial Infeasibility

  32. Minimum-degree Partition Nullstellensatz Certificates � �� � n Let S n i.e., | S n k denote the set of k -subsets of { 1 , . . . , n } k | = k Susan Margulies, US Naval Academy NulLA and Combinatorial Infeasibility

  33. Minimum-degree Partition Nullstellensatz Certificates � �� � n Let S n i.e., | S n k denote the set of k -subsets of { 1 , . . . , n } k | = k Theorem (S.M., S. Onn, 2012) Given a set of non-partitionable integers W = { w 1 , . . . , w n } encoded as a system of polynomial equations as above, there exists a minimum-degree Nullstellensatz certificate for the non-existence of a partition of W as follows: n n � � c i , s x s � � � b s x s �� � � � � � ( x 2 1 = i − 1) + w i x i . s ∈ S n i =1 s ∈ S n \ i i =1 k even k odd k k ≤ n − 1 k ≤ n k Moreover, every Nullstellensatz certificate associated with the above system of polynomial equations contains exactly one monomial for each of the even parity subsets of S n \ i , and exactly k one monomial for each of the odd parity subsets of S n k . Susan Margulies, US Naval Academy NulLA and Combinatorial Infeasibility

  34. Minimum-degree Partition Nullstellensatz Certificates � �� � n Let S n i.e., | S n k denote the set of k -subsets of { 1 , . . . , n } k | = k Theorem (S.M., S. Onn, 2012) Given a set of non-partitionable integers W = { w 1 , . . . , w n } encoded as a system of polynomial equations as above, there exists a minimum-degree Nullstellensatz certificate for the non-existence of a partition of W as follows: n n � � c i , s x s � � � b s x s �� � � � � � ( x 2 1 = i − 1) + w i x i . s ∈ S n i =1 s ∈ S n \ i i =1 k even k odd k k ≤ n − 1 k ≤ n k Moreover, every Nullstellensatz certificate associated with the above system of polynomial equations contains exactly one monomial for each of the even parity subsets of S n \ i , and exactly k one monomial for each of the odd parity subsets of S n k . Susan Margulies, US Naval Academy NulLA and Combinatorial Infeasibility

  35. Minimum-degree Partition Nullstellensatz Certificates � �� � n Let S n i.e., | S n k denote the set of k -subsets of { 1 , . . . , n } k | = k Theorem (S.M., S. Onn, 2012) Given a set of non-partitionable integers W = { w 1 , . . . , w n } encoded as a system of polynomial equations as above, there exists a minimum-degree Nullstellensatz certificate for the non-existence of a partition of W as follows: n n � � c i , s x s � � � b s x s �� � � � � � ( x 2 1 = i − 1) + w i x i . s ∈ S n i =1 s ∈ S n \ i i =1 k even k odd k k ≤ n − 1 k ≤ n k Moreover, every Nullstellensatz certificate associated with the above system of polynomial equations contains exactly one monomial for each of the even parity subsets of S n \ i , and exactly k one monomial for each of the odd parity subsets of S n k . Note: degree is n for n odd and n − 1 for n even. Susan Margulies, US Naval Academy NulLA and Combinatorial Infeasibility

  36. Minimum-degree Partition Nullstellensatz Certificates � �� � n Let S n i.e., | S n k denote the set of k -subsets of { 1 , . . . , n } k | = k Theorem (S.M., S. Onn, 2012) Given a set of non-partitionable integers W = { w 1 , . . . , w n } encoded as a system of polynomial equations as above, there exists a minimum-degree Nullstellensatz certificate for the non-existence of a partition of W as follows: n n � � c i , s x s � � � b s x s �� � � � � � ( x 2 1 = i − 1) + w i x i . s ∈ S n i =1 s ∈ S n \ i i =1 k even k odd k k ≤ n − 1 k ≤ n k Moreover, every Nullstellensatz certificate associated with the above system of polynomial equations contains exactly one monomial for each of the even parity subsets of S n \ i , and exactly k one monomial for each of the odd parity subsets of S n k . Note: degree is n for n odd and n − 1 for n even. Susan Margulies, US Naval Academy NulLA and Combinatorial Infeasibility

  37. Minimum-degree Partition Nullstellensatz Certificates � �� � n Let S n i.e., | S n k denote the set of k -subsets of { 1 , . . . , n } k | = k Theorem (S.M., S. Onn, 2012) Given a set of non-partitionable integers W = { w 1 , . . . , w n } encoded as a system of polynomial equations as above, there exists a minimum-degree Nullstellensatz certificate for the non-existence of a partition of W as follows: n n � � c i , s x s � � � b s x s �� � � � � � ( x 2 1 = i − 1) + w i x i . s ∈ S n i =1 s ∈ S n \ i i =1 k even k odd k k ≤ n − 1 k ≤ n k Moreover, every Nullstellensatz certificate associated with the above system of polynomial equations contains exactly one monomial for each of the even parity subsets of S n \ i , and exactly k one monomial for each of the odd parity subsets of S n k . Note: degree is n for n odd and n − 1 for n even. Susan Margulies, US Naval Academy NulLA and Combinatorial Infeasibility

  38. Minimum-degree Partition Nullstellensatz Certificates � �� � n Let S n i.e., | S n k denote the set of k -subsets of { 1 , . . . , n } k | = k Theorem (S.M., S. Onn, 2012) Given a set of non-partitionable integers W = { w 1 , . . . , w n } encoded as a system of polynomial equations as above, there exists a minimum-degree Nullstellensatz certificate for the non-existence of a partition of W as follows: n n � � c i , s x s � � � b s x s �� � � � � � ( x 2 1 = i − 1) + w i x i . s ∈ S n i =1 s ∈ S n \ i i =1 k even k odd k k ≤ n − 1 k ≤ n k Moreover, every Nullstellensatz certificate associated with the above system of polynomial equations contains exactly one monomial for each of the even parity subsets of S n \ i , and exactly k one monomial for each of the odd parity subsets of S n k . Note: degree is n for n odd and n − 1 for n even. Susan Margulies, US Naval Academy NulLA and Combinatorial Infeasibility

  39. Minimum-degree Partition Nullstellensatz Certificates � �� � n Let S n i.e., | S n k denote the set of k -subsets of { 1 , . . . , n } k | = k Theorem (S.M., S. Onn, 2012) Given a set of non-partitionable integers W = { w 1 , . . . , w n } encoded as a system of polynomial equations as above, there exists a minimum-degree Nullstellensatz certificate for the non-existence of a partition of W as follows: n n � � c i , s x s � � � b s x s �� � � � � � ( x 2 1 = i − 1) + w i x i . s ∈ S n i =1 s ∈ S n \ i i =1 k even k odd k k ≤ n − 1 k ≤ n k Moreover, every Nullstellensatz certificate associated with the above system of polynomial equations contains exactly one monomial for each of the even parity subsets of S n \ i , and exactly k one monomial for each of the odd parity subsets of S n k . Note: degree is n for n odd and n − 1 for n even. Susan Margulies, US Naval Academy NulLA and Combinatorial Infeasibility

  40. Minimum-degree Partition Nullstellensatz Certificates � �� � n Let S n i.e., | S n k denote the set of k -subsets of { 1 , . . . , n } k | = k Theorem (S.M., S. Onn, 2012) Given a set of non-partitionable integers W = { w 1 , . . . , w n } encoded as a system of polynomial equations as above, there exists a minimum-degree Nullstellensatz certificate for the non-existence of a partition of W as follows: n n � � c i , s x s � � � b s x s �� � � � � � ( x 2 1 = i − 1) + w i x i . s ∈ S n i =1 s ∈ S n \ i i =1 k even k odd k k ≤ n − 1 k ≤ n k Moreover, every Nullstellensatz certificate associated with the above system of polynomial equations contains exactly one monomial for each of the even parity subsets of S n \ i , and exactly k one monomial for each of the odd parity subsets of S n k . Note: degree is n for n odd and n − 1 for n even. Susan Margulies, US Naval Academy NulLA and Combinatorial Infeasibility

  41. Minimum-degree Partition Nullstellensatz Certificates � �� � n Let S n i.e., | S n k denote the set of k -subsets of { 1 , . . . , n } k | = k Theorem (S.M., S. Onn, 2012) Given a set of non-partitionable integers W = { w 1 , . . . , w n } encoded as a system of polynomial equations as above, there exists a minimum-degree Nullstellensatz certificate for the non-existence of a partition of W as follows: n n � � c i , s x s � � � b s x s �� � � � � � ( x 2 1 = i − 1) + w i x i . s ∈ S n i =1 s ∈ S n \ i i =1 k even k odd k k ≤ n − 1 k ≤ n k Moreover, every Nullstellensatz certificate associated with the above system of polynomial equations contains exactly one monomial for each of the even parity subsets of S n \ i , and exactly k one monomial for each of the odd parity subsets of S n k . Note: degree is n for n odd and n − 1 for n even. Susan Margulies, US Naval Academy NulLA and Combinatorial Infeasibility

  42. Minimum-degree Partition Nullstellensatz Certificates � �� � n Let S n i.e., | S n k denote the set of k -subsets of { 1 , . . . , n } k | = k Theorem (S.M., S. Onn, 2012) Given a set of non-partitionable integers W = { w 1 , . . . , w n } encoded as a system of polynomial equations as above, there exists a minimum-degree Nullstellensatz certificate for the non-existence of a partition of W as follows: n n � � c i , s x s � � � b s x s �� � � � � � ( x 2 1 = i − 1) + w i x i . s ∈ S n i =1 s ∈ S n \ i i =1 k even k odd k k ≤ n − 1 k ≤ n k Moreover, every Nullstellensatz certificate associated with the above system of polynomial equations contains exactly one monomial for each of the even parity subsets of S n \ i , and exactly k one monomial for each of the odd parity subsets of S n k . Note: certificate is both high degree and dense. Susan Margulies, US Naval Academy NulLA and Combinatorial Infeasibility

  43. Minimum-degree Nullstellensatz Certificates Example Question: Let W = { 1 , 3 , 5 , 2 } . Is W partitionable? x 2 x 2 x 3 x 2 1 − 1 = 0 , 2 − 1 = 0 , 3 − 1 = 0 , 4 − 1 = 0 , x 1 + 3 x 2 + 5 x 3 + 2 x 4 = 0 . Susan Margulies, US Naval Academy NulLA and Combinatorial Infeasibility

  44. Minimum-degree Nullstellensatz Certificates Example Question: Let W = { 1 , 3 , 5 , 2 } . Is W partitionable? Answer: No! x 2 x 2 x 3 x 2 1 − 1 = 0 , 2 − 1 = 0 , 3 − 1 = 0 , 4 − 1 = 0 , x 1 + 3 x 2 + 5 x 3 + 2 x 4 = 0 . � � − 155 693 + 842 3465 x 2 x 3 − 188 693 x 2 x 4 + 908 ( x 2 1 = 1 − 1 ) 3465 x 3 x 4 � � 231 + 842 1 1155 x 1 x 3 − 188 231 x 1 x 4 + 292 ( x 2 + − 1155 x 3 x 4 2 − 1 ) � � − 467 693 + 842 693 x 1 x 2 + 908 693 x 1 x 4 + 292 ( x 2 + 693 x 2 x 4 3 − 1 ) � � − 68 693 − 376 693 x 1 x 2 + 1816 3465 x 1 x 3 + 584 ( x 2 + 3465 x 2 x 3 4 − 1 ) � 155 693 x 2 + 467 1 3465 x 3 + 34 693 x 4 − 842 + 693 x 1 + 3465 x 1 x 2 x 3 � + 188 693 x 1 x 2 x 4 − 908 3465 x 1 x 3 x 4 − 292 3465 x 2 x 3 x 4 ( x 1 + 3x 2 + 5x 3 + 2x 4 ) . Susan Margulies, US Naval Academy NulLA and Combinatorial Infeasibility

  45. Minimum-degree Nullstellensatz Certificates Example Question: Let W = { 1 , 3 , 5 , 2 } . Is W partitionable? Answer: No! x 2 x 2 x 3 x 2 1 − 1 = 0 , 2 − 1 = 0 , 3 − 1 = 0 , 4 − 1 = 0 , x 1 + 3 x 2 + 5 x 3 + 2 x 4 = 0 . � � − 155 693 + 842 3465 x 2 x 3 − 188 693 x 2 x 4 + 908 ( x 2 1 = 1 − 1 ) 3465 x 3 x 4 � � 231 + 842 1 1155 x 1 x 3 − 188 231 x 1 x 4 + 292 ( x 2 + − 1155 x 3 x 4 2 − 1 ) � � − 467 693 + 842 693 x 1 x 2 + 908 693 x 1 x 4 + 292 ( x 2 + 693 x 2 x 4 3 − 1 ) � � − 68 693 − 376 693 x 1 x 2 + 1816 3465 x 1 x 3 + 584 ( x 2 + 3465 x 2 x 3 4 − 1 ) � 155 693 x 2 + 467 1 3465 x 3 + 34 693 x 4 − 842 + 693 x 1 + 3465 x 1 x 2 x 3 � + 188 693 x 1 x 2 x 4 − 908 3465 x 1 x 3 x 4 − 292 3465 x 2 x 3 x 4 ( x 1 + 3x 2 + 5x 3 + 2x 4 ) . Susan Margulies, US Naval Academy NulLA and Combinatorial Infeasibility

  46. The Partition Matrix: Extract a Square Linear System Let W = { w 1 , w 2 , w 3 } .   0 w 3 w 2 w 1 w 2 w 3 0 w 1     0 w 1 w 3 w 2   0 w 1 w 2 w 3 Susan Margulies, US Naval Academy NulLA and Combinatorial Infeasibility

  47. The Partition Matrix: Extract a Square Linear System Let W = { w 1 , w 2 , w 3 } . w 3   0 w 3 w 2 w 1 w 3 w 2 w 3 0 w 1   w 3   0 w 1 w 3 w 2   w 3 0 w 1 w 2 w 3 Susan Margulies, US Naval Academy NulLA and Combinatorial Infeasibility

  48. The Partition Matrix: Extract a Square Linear System Let W = { w 1 , w 2 , w 3 } . w 1 w 2 w 3   0 w 3 w 2 w 1 w 3 w 2 w 3 0 w 1   w 3   0 w 1 w 3 w 2   w 3 0 w 1 w 2 w 3 Susan Margulies, US Naval Academy NulLA and Combinatorial Infeasibility

  49. The Partition Matrix: Extract a Square Linear System Let W = { w 1 , w 2 , w 3 } . w 1 w 2 w 3   0 w 3 w 2 w 1 w 1 w 2 w 3 w 2 w 3 0 w 1   w 3   0 w 1 w 3 w 2   w 3 0 w 1 w 2 w 3 Susan Margulies, US Naval Academy NulLA and Combinatorial Infeasibility

  50. The Partition Matrix: Extract a Square Linear System Let W = { w 1 , w 2 , w 3 } . w 1 w 2 w 3   0 w 3 w 2 w 1 w 1 w 2 w 3 w 2 w 3 0 w 1   w 2 w 1 w 3   0 w 1 w 3 w 2   w 3 0 w 1 w 2 w 3 Susan Margulies, US Naval Academy NulLA and Combinatorial Infeasibility

  51. The Partition Matrix: Extract a Square Linear System Let W = { w 1 , w 2 , w 3 } . w 1 w 2 w 3   0 w 3 w 2 w 1 w 1 w 2 w 3 w 2 w 3 0 w 1   w 2 w 1 w 3   0 w 1 w 3 w 2   w 1 w 2 w 3 0 w 1 w 2 w 3 Susan Margulies, US Naval Academy NulLA and Combinatorial Infeasibility

  52. The Partition Matrix: Extract a Square Linear System Let W = { w 1 , w 2 , w 3 } . − + w 1 + w 2 + w 3   0 w 3 w 2 w 1 − w 1 + w 2 + w 3 w 2 w 3 0 w 1   − w 2 + w 1 + w 3   0 w 1 w 3 w 2   − w 1 − w 2 + w 3 0 w 1 w 2 w 3 Susan Margulies, US Naval Academy NulLA and Combinatorial Infeasibility

  53. The Partition Matrix: Extract a Square Linear System Let W = { w 1 , w 2 , w 3 } . − + w 1 + w 2 + w 3   0 w 3 w 2 w 1 − w 1 + w 2 + w 3 w 2 w 3 0 w 1   − w 2 + w 1 + w 3   0 w 1 w 3 w 2   − w 1 − w 2 + w 3 0 w 1 w 2 w 3 ( w 1 + w 2 + w 3 )( − w 1 + w 2 + w 3 )( w 1 − w 2 + w 3 )( − w 1 − w 2 + w 3 ) � �� � partition polynomial Susan Margulies, US Naval Academy NulLA and Combinatorial Infeasibility

  54. The Partition Matrix: Extract a Square Linear System Let W = { w 1 , w 2 , w 3 } . − + w 1 + w 2 + w 3   0 w 3 w 2 w 1 − w 1 + w 2 + w 3 w 2 w 3 0 w 1   − w 2 + w 1 + w 3   0 w 1 w 3 w 2   − w 1 − w 2 + w 3 0 w 1 w 2 w 3 ( w 1 + w 2 + w 3 )( − w 1 + w 2 + w 3 )( w 1 − w 2 + w 3 )( − w 1 − w 2 + w 3 ) � �� � partition polynomial Susan Margulies, US Naval Academy NulLA and Combinatorial Infeasibility

  55. The Partition Matrix: Extract a Square Linear System Let W = { w 1 , w 2 , w 3 } . − + w 1 + w 2 + w 3   0 w 3 w 2 w 1 − w 1 + w 2 + w 3 w 2 w 3 0 w 1   − w 2 + w 1 + w 3   0 w 1 w 3 w 2   − w 1 − w 2 + w 3 0 w 1 w 2 w 3 The determinant of the above partition matrix is the ( w 1 + w 2 + w 3 )( − w 1 + w 2 + w 3 )( w 1 − w 2 + w 3 )( − w 1 − w 2 + w 3 ) � �� � partition polynomial Susan Margulies, US Naval Academy NulLA and Combinatorial Infeasibility

  56. Another Example of the Partition Matrix Let W = { w 1 , . . . , w 4 } . The partition matrix P is   w 4 w 3 w 2 w 1 0 0 0 0 0 0 0 0 w 3 w 4 w 2 w 1     w 2 0 w 4 0 w 3 0 w 1 0     w 1 0 0 w 4 0 w 3 w 2 0   P = ,   0 w 2 w 3 0 w 4 0 0 w 1     0 w 1 0 w 3 0 w 4 0 w 2     0 0 0 0  w 1 w 2 w 4 w 3  0 0 0 0 w 1 w 2 w 3 w 4 Susan Margulies, US Naval Academy NulLA and Combinatorial Infeasibility

  57. Another Example of the Partition Matrix Let W = { w 1 , . . . , w 4 } . The partition matrix P is   w 4 w 3 w 2 w 1 0 0 0 0 0 0 0 0 w 3 w 4 w 2 w 1     w 2 0 w 4 0 w 3 0 w 1 0     w 1 0 0 w 4 0 w 3 w 2 0   P = ,   0 w 2 w 3 0 w 4 0 0 w 1     0 w 1 0 w 3 0 w 4 0 w 2     0 0 0 0  w 1 w 2 w 4 w 3  0 0 0 0 w 1 w 2 w 3 w 4 det( P ) = ( w 1 + w 2 + w 3 + w 4 )( − w 1 + w 2 + w 3 + w 4 )( w 1 − w 2 + w 3 + w 4 ) ( w 1 + w 2 − w 3 + w 4 )( − w 1 + w 2 − w 3 + w 4 )( − w 1 − w 2 + w 3 + w 4 ) ( w 1 − w 2 − w 3 + w 4 )( − w 1 − w 2 − w 3 + w 4 ) . “partition polynomial” Susan Margulies, US Naval Academy NulLA and Combinatorial Infeasibility

  58. Determinant and Partition Polynomial Theorem (S.M., S. Onn, 2012) The determinant of the partition matrix is the partition polynomial. Susan Margulies, US Naval Academy NulLA and Combinatorial Infeasibility

  59. Hilbert’s Nullstellensatz Numeric Coefficients and the Partition Polynomial Given a square non-singular matrix A , Cramer’s rule states that Ax = b can be solved according to the formula x i = det( A | i b ) , det( A ) where A | i b is the matrix A with the i -th column replaced with the right-hand side vector b . Susan Margulies, US Naval Academy NulLA and Combinatorial Infeasibility

  60. Recall the non-partitionable W = { 1 , 3 , 5 , 2 } : � � − 155 693 + 842 3465 x 2 x 3 − 188 693 x 2 x 4 + 908 ( x 2 1 = 3465 x 3 x 4 1 − 1 ) � � 231 + 842 1 1155 x 1 x 3 − 188 231 x 1 x 4 + 292 ( x 2 + − 2 − 1 ) 1155 x 3 x 4 � � − 467 693 + 842 693 x 1 x 2 + 908 693 x 1 x 4 + 292 ( x 2 + 693 x 2 x 4 3 − 1 ) � � − 68 693 − 376 693 x 1 x 2 + 1816 3465 x 1 x 3 + 584 ( x 2 + 4 − 1 ) 3465 x 2 x 3 � 155 693 x 2 + 467 1 3465 x 3 + 34 693 x 4 − 842 + 693 x 1 + 3465 x 1 x 2 x 3 � + 188 693 x 1 x 2 x 4 − 908 3465 x 1 x 3 x 4 − 292 ( x 1 + 3x 2 + 5x 3 + 2x 4 ) . 3465 x 2 x 3 x 4 Susan Margulies, US Naval Academy NulLA and Combinatorial Infeasibility

  61. Recall the non-partitionable W = { 1 , 3 , 5 , 2 } : � � − 155 693 + 842 3465 x 2 x 3 − 188 693 x 2 x 4 + 908 ( x 2 1 = 3465 x 3 x 4 1 − 1 ) � � 231 + 842 1 1155 x 1 x 3 − 188 231 x 1 x 4 + 292 ( x 2 + − 2 − 1 ) 1155 x 3 x 4 � � − 467 693 + 842 693 x 1 x 2 + 908 693 x 1 x 4 + 292 ( x 2 + 693 x 2 x 4 3 − 1 ) � � − 68 693 − 376 693 x 1 x 2 + 1816 3465 x 1 x 3 + 584 ( x 2 + 4 − 1 ) 3465 x 2 x 3 � 155 693 x 2 + 467 1 3465 x 3 + 34 693 x 4 − 842 + 693 x 1 + 3465 x 1 x 2 x 3 � + 188 693 x 1 x 2 x 4 − 908 3465 x 1 x 3 x 4 − 292 ( x 1 + 3x 2 + 5x 3 + 2x 4 ) . 3465 x 2 x 3 x 4 Susan Margulies, US Naval Academy NulLA and Combinatorial Infeasibility

  62. Recall the non-partitionable W = { 1 , 3 , 5 , 2 } : � � − 155 693 + 842 3465 x 2 x 3 − 188 693 x 2 x 4 + 908 ( x 2 1 = 3465 x 3 x 4 1 − 1 ) � � 231 + 842 1 1155 x 1 x 3 − 188 231 x 1 x 4 + 292 ( x 2 + − 2 − 1 ) 1155 x 3 x 4 � � − 467 693 + 842 693 x 1 x 2 + 908 693 x 1 x 4 + 292 ( x 2 + 693 x 2 x 4 3 − 1 ) � � − 68 693 − 376 693 x 1 x 2 + 1816 3465 x 1 x 3 + 584 ( x 2 + 4 − 1 ) 3465 x 2 x 3 � 155 693 x 2 + 467 1 3465 x 3 + 34 693 x 4 − 842 + 693 x 1 + 3465 x 1 x 2 x 3 � + 188 693 x 1 x 2 x 4 − 908 3465 x 1 x 3 x 4 − 292 ( x 1 + 3x 2 + 5x 3 + 2x 4 ) . 3465 x 2 x 3 x 4 − 51975 = (1 + 3 + 5 + 2)( − 1 + 3 + 5 + 2)(1 − 3 + 5 + 2)(1 + 3 − 5 + 2) ( − 1 − 3 + 5 + 2)( − 1 + 3 − 5 + 2)(1 − 3 − 5 + 2)( − 1 − 3 − 5 + 2) . Susan Margulies, US Naval Academy NulLA and Combinatorial Infeasibility

  63. Recall the non-partitionable W = { 1 , 3 , 5 , 2 } : � � − 155 693 + 842 3465 x 2 x 3 − 188 693 x 2 x 4 + 908 ( x 2 1 = 3465 x 3 x 4 1 − 1 ) � � 231 + 842 1 1155 x 1 x 3 − 188 231 x 1 x 4 + 292 ( x 2 + − 2 − 1 ) 1155 x 3 x 4 � � − 467 693 + 842 693 x 1 x 2 + 908 693 x 1 x 4 + 292 ( x 2 + 693 x 2 x 4 3 − 1 ) � � − 68 693 − 376 693 x 1 x 2 + 1816 3465 x 1 x 3 + 584 ( x 2 + 4 − 1 ) 3465 x 2 x 3 � 155 693 x 2 + 467 1 3465 x 3 + 34 693 x 4 − 842 + 693 x 1 + 3465 x 1 x 2 x 3 � + 188 693 x 1 x 2 x 4 − 908 3465 x 1 x 3 x 4 − 292 ( x 1 + 3x 2 + 5x 3 + 2x 4 ) . 3465 x 2 x 3 x 4 − 51975 = (1 + 3 + 5 + 2)( − 1 + 3 + 5 + 2)(1 − 3 + 5 + 2)(1 + 3 − 5 + 2) ( − 1 − 3 + 5 + 2)( − 1 + 3 − 5 + 2)(1 − 3 − 5 + 2)( − 1 − 3 − 5 + 2) . Via Cramer’s rule, we see that the unknown b 4 is equal to b 4 = − 2550 − 51975 Susan Margulies, US Naval Academy NulLA and Combinatorial Infeasibility

  64. Recall the non-partitionable W = { 1 , 3 , 5 , 2 } : � � − 155 693 + 842 3465 x 2 x 3 − 188 693 x 2 x 4 + 908 ( x 2 1 = 3465 x 3 x 4 1 − 1 ) � � 231 + 842 1 1155 x 1 x 3 − 188 231 x 1 x 4 + 292 ( x 2 + − 2 − 1 ) 1155 x 3 x 4 � � − 467 693 + 842 693 x 1 x 2 + 908 693 x 1 x 4 + 292 ( x 2 + 693 x 2 x 4 3 − 1 ) � � − 68 693 − 376 693 x 1 x 2 + 1816 3465 x 1 x 3 + 584 ( x 2 + 4 − 1 ) 3465 x 2 x 3 � 155 693 x 2 + 467 1 3465 x 3 + 34 693 x 4 − 842 + 693 x 1 + 3465 x 1 x 2 x 3 � + 188 693 x 1 x 2 x 4 − 908 3465 x 1 x 3 x 4 − 292 ( x 1 + 3x 2 + 5x 3 + 2x 4 ) . 3465 x 2 x 3 x 4 − 51975 = (1 + 3 + 5 + 2)( − 1 + 3 + 5 + 2)(1 − 3 + 5 + 2)(1 + 3 − 5 + 2) ( − 1 − 3 + 5 + 2)( − 1 + 3 − 5 + 2)(1 − 3 − 5 + 2)( − 1 − 3 − 5 + 2) . Via Cramer’s rule, we see that the unknown b 4 is equal to b 4 = − 2550 − 51975 = 34 693 . Susan Margulies, US Naval Academy NulLA and Combinatorial Infeasibility

  65. Definition of Graph Coloring Graph coloring: Given a graph G , and an integer k , can the vertices be colored with k colors in such a way that no two adjacent vertices are the same color? Susan Margulies, US Naval Academy NulLA and Combinatorial Infeasibility

  66. Definition of Graph Coloring Graph coloring: Given a graph G , and an integer k , can the vertices be colored with k colors in such a way that no two adjacent vertices are the same color? Petersen Graph: 3-colorable Susan Margulies, US Naval Academy NulLA and Combinatorial Infeasibility

  67. Graph 3-Coloring as a System of Polynomial Equations over C (D. Bayer) one variable per vertex : x 1 , . . . , x n Susan Margulies, US Naval Academy NulLA and Combinatorial Infeasibility

  68. Graph 3-Coloring as a System of Polynomial Equations over C (D. Bayer) one variable per vertex : x 1 , . . . , x n vertex polynomials: For every vertex i = 1 , . . . , n , x 3 i − 1 = 0 Susan Margulies, US Naval Academy NulLA and Combinatorial Infeasibility

  69. Graph 3-Coloring as a System of Polynomial Equations over C (D. Bayer) one variable per vertex : x 1 , . . . , x n vertex polynomials: For every vertex i = 1 , . . . , n , x 3 i − 1 = 0 edge polynomials: For every edge ( i , j ) ∈ E ( G ), x 2 i + x i x j + x 2 j = 0 Susan Margulies, US Naval Academy NulLA and Combinatorial Infeasibility

  70. Graph 3-Coloring as a System of Polynomial Equations over C (D. Bayer) one variable per vertex : x 1 , . . . , x n vertex polynomials: For every vertex i = 1 , . . . , n , x 3 i − 1 = 0 edge polynomials: For every edge ( i , j ) ∈ E ( G ), x 3 i − x 3 j = x 2 i + x i x j + x 2 j = 0 x i − x j Susan Margulies, US Naval Academy NulLA and Combinatorial Infeasibility

  71. Graph 3-Coloring as a System of Polynomial Equations over C (D. Bayer) one variable per vertex : x 1 , . . . , x n vertex polynomials: For every vertex i = 1 , . . . , n , x 3 i − 1 = 0 edge polynomials: For every edge ( i , j ) ∈ E ( G ), x 3 i − x 3 j = x 2 i + x i x j + x 2 j = 0 x i − x j Theorem: Let G be a graph encoded as the above ( n + m ) system of equations. Then this system has a solution if and only if G is 3-colorable. Susan Margulies, US Naval Academy NulLA and Combinatorial Infeasibility

  72. Petersen Graph = ⇒ System of Polynomial Equations Figure: Is the Petersen graph 3-colorable? x 3 0 − 1 = 0 , x 3 x 2 0 + x 0 x 1 + x 2 1 = 0 , x 2 0 + x 0 x 4 + x 2 1 − 1 = 0 , 4 = 0 x 3 2 − 1 = 0 , x 3 x 2 0 + x 0 x 5 + x 2 5 = 0 , x 2 1 + x 1 x 2 + x 2 3 − 1 = 0 , 2 = 0 x 3 4 − 1 = 0 , x 3 x 2 1 + x 1 x 6 + x 2 6 = 0 , x 2 2 + x 2 x 3 + x 2 5 − 1 = 0 , 3 = 0 x 3 6 − 1 = 0 , x 3 7 − 1 = 0 , · · · · · · · · · · · · x 3 8 − 1 = 0 , x 3 x 2 6 + x 6 x 8 + x 2 8 = 0 , x 2 7 + x 7 x 9 + x 2 9 − 1 = 0 , 9 = 0 Susan Margulies, US Naval Academy NulLA and Combinatorial Infeasibility

  73. Where is the Infinite Family of Graphs that Grow over C ? 4 Susan Margulies, US Naval Academy NulLA and Combinatorial Infeasibility

  74. Where is the Infinite Family of Graphs that Grow over C ? 4 Flower, Kneser, Gr¨ otzsch, Jin, Mycielski graphs have degree 4. Susan Margulies, US Naval Academy NulLA and Combinatorial Infeasibility

  75. Where is the Infinite Family of Graphs that Grow over C ? 4 Flower, Kneser, Gr¨ otzsch, Jin, Mycielski graphs have degree 4. Theorem: Every Nullstellensatz certificate of a non-3-colorable graph has degree at least four. Susan Margulies, US Naval Academy NulLA and Combinatorial Infeasibility

  76. Where is the Infinite Family of Graphs that Grow over C ? 4 Flower, Kneser, Gr¨ otzsch, Jin, Mycielski graphs have degree 4. Theorem: Every Nullstellensatz certificate of a non-3-colorable graph has degree at least four. Theorem: For n ≥ 4, a minimum-degree Nullstellensatz certificate of non-3-colorability for cliques and odd wheels has degree exactly four. Susan Margulies, US Naval Academy NulLA and Combinatorial Infeasibility

  77. Graph 3-Coloring as a System of Polynomial Equations over F 2 (inspired by Bayer) one variable per vertex : x 1 , . . . , x n vertex polynomials: For every vertex i = 1 , . . . , n , x 3 i + 1 = 0 edge polynomials: For every edge ( i , j ) ∈ E ( G ), x 2 i + x i x j + x 2 j = 0 Theorem: Let G be a graph encoded as the above ( n + m ) system of equations. Then this system has a solution if and only if G is 3-colorable. Susan Margulies, US Naval Academy NulLA and Combinatorial Infeasibility

  78. Where is the Infinite Family of Graphs that Grow over F 2 ? 1 Susan Margulies, US Naval Academy NulLA and Combinatorial Infeasibility

  79. Where is the Infinite Family of Graphs that Grow over F 2 ? 1 Theorem: Every Nullstellensatz certificate of a non-3-colorable graph has degree at least one. Susan Margulies, US Naval Academy NulLA and Combinatorial Infeasibility

  80. Where is the Infinite Family of Graphs that Grow over F 2 ? 1 Theorem: Every Nullstellensatz certificate of a non-3-colorable graph has degree at least one. Theorem: For n ≥ 4, a minimum-degree Nullstellensatz certificate of non-3-colorability for cliques and odd wheels has degree exactly one. Susan Margulies, US Naval Academy NulLA and Combinatorial Infeasibility

  81. Experimental results for NulLA 3-colorability Graph vertices edges rows cols deg sec Mycielski 7 95 755 64,281 71,726 Mycielski 9 383 7,271 2,477,931 2,784,794 Mycielski 10 767 22,196 15,270,943 17,024,333 (8 , 3)-Kneser 56 280 15,737 15,681 (10 , 4)-Kneser 210 1,575 349,651 330,751 (12 , 5)-Kneser 792 8,316 7,030,585 6,586,273 (13 , 5)-Kneser 1,287 36,036 45,980,650 46,378,333 1-Insertions 5 202 1,227 268,049 247,855 2-Insertions 5 597 3,936 2,628,805 2,349,793 3-Insertions 5 1,406 9,695 15,392,209 13,631,171 ash331GPIA 662 4,185 3,147,007 2,770,471 ash608GPIA 1,216 7,844 10,904,642 9,538,305 ash958GPIA 1,916 12,506 27,450,965 23,961,497 Table: Graphs without 4-cliques. Susan Margulies, US Naval Academy NulLA and Combinatorial Infeasibility

  82. Experimental results for NulLA 3-colorability Graph vertices edges rows cols deg sec Mycielski 7 95 755 64,281 71,726 1 Mycielski 9 383 7,271 2,477,931 2,784,794 1 Mycielski 10 767 22,196 15,270,943 17,024,333 1 (8 , 3)-Kneser 56 280 15,737 15,681 1 (10 , 4)-Kneser 210 1,575 349,651 330,751 1 (12 , 5)-Kneser 792 8,316 7,030,585 6,586,273 1 (13 , 5)-Kneser 1,287 36,036 45,980,650 46,378,333 1 1-Insertions 5 202 1,227 268,049 247,855 1 2-Insertions 5 597 3,936 2,628,805 2,349,793 1 3-Insertions 5 1,406 9,695 15,392,209 13,631,171 1 ash331GPIA 662 4,185 3,147,007 2,770,471 1 ash608GPIA 1,216 7,844 10,904,642 9,538,305 1 ash958GPIA 1,916 12,506 27,450,965 23,961,497 1 Table: Graphs without 4-cliques. Susan Margulies, US Naval Academy NulLA and Combinatorial Infeasibility

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