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The Importance of Being Zero T OM AS R ECIO , J. R AFAEL S ENDRA AND - PowerPoint PPT Presentation

S1 ISSAC18, July 16-19, 2018, New York, NY, USA. The Importance of Being Zero T OM AS R ECIO , J. R AFAEL S ENDRA AND C ARLOS V ILLARINO . P RESENTED BY : L AUREANO G ONZ ALEZ -V EGA . T HIS WORK IS PARTIALLY SUPPORTED BY THE RESEARCH


  1. S1 ISSAC’18, July 16-19, 2018, New York, NY, USA. The Importance of Being Zero T OM ´ AS R ECIO , J. R AFAEL S ENDRA AND C ARLOS V ILLARINO . P RESENTED BY : L AUREANO G ONZ ´ ALEZ -V EGA . T HIS WORK IS PARTIALLY SUPPORTED BY THE RESEARCH PROJECT MTM2017-88796-P (S PANISH M INISTERIO DE E CONOM ´ IA Y C OMPETITIVIDAD ).

  2. S2 Introduction: The importance of Being Zero. This paper is related with the classical problem of determining if a polynomial (in our case, an ideal I) is or not the zero polynomial (or zero ideal). • Polynomial zero testing: P ( x ) ∈ K [ x ] − → P ( x ) ≡ 0 ? deg( P ( x )) ≤ d • by evaluation on a finite number of instances. We assume that we only know some very limited data: number of variables and an upper bound for the geometric degree (in the sense of Heintz) gdeg( V (I)) . And we want to accomplish the zero test just by means of an oracle that allows us to check, given a point in K n , whether this point is or not in V (I) .

  3. S3 Introduction: The Importance of Being Zero. • For checking if x 2 − 1 � x 3 � � � = x 2 + x − 1 � � � � � � 1 x + 1 � � or equivalently cheking whether the polynomial P ( x ) x 2 − 1 � � x 3 � � − ( x 2 + x − 1) � � P ( x ) = � � � � 1 x + 1 � � is zero, we evaluate on a finite number of instances, • Taking into account that the determinant is a polynomial of degree bounded by d = 3 • We evaluate P ( x ) in d + 1 different point of K – For x ∈ { 0 , 1 , 2 , 3 } , P ( x ) = 0 and hence P ( x ) ≡ 0 .

  4. S4 Introduction: The Importance of Being Zero. In this paper we extend this technique P ( x ) ∈ K [ x ] − → I ⊂ K [x 1 , . . . , x n ] deg( P ( x )) ≤ d − → gdeg( V (I)) d + 1 points − → Test set and focus our attention on these goals: • Goal I: zero testing of an ideal, • Goal II: zero testing elimination ideals, • Goal III: application of this technique to automated theorem proving.

  5. S5 Goal I: Zero Testing of an Ideal. • For a polynomial ideal I – with coefficients in K (field alg. closed of characteristic zero), – in n-variables { x 1 , . . . , x n } , – gdeg( V (I)) ≤ d • we present an algorithm for deciding whether I = < 0 > or I � = < 0 > or equivalently whether V (I) = K n or V (I) � = K n . • performing in a finite number of instances (Test Set!!) whether a point is or not in V (I) (Oracle !!)

  6. S6 Goal II: Zero Testing Elimination Ideals. • For a polynomial ideal I ⊂ K [x 1 , . . . , x n ] where – gdeg( V (I)) ≤ d , • Let I r = I ∩ K [x 1 , . . . , x r ] be the r -elimination ideal of I . • We present an algorithm for deciding whether I r = < 0 > or I r � = < 0 > by instances (not directly using symbolic techniques). – We are developing the theoretical framework in this context. – We are applying this technique to automated reasoning for geometric statements. – Implementation in GeoGebra 1 (in process....) by other people. 1 E.g. ISSAC 2016 Software Demo Award: Development of automatic reasoning tools in GeoGebra. Miguel Ab´ anades, Francisco Botana, Zolt´ an Kov´ acs, Tom´ as Recio, Csilla S´ olyom-Gecse.

  7. S7 Goal III: Application of this Technique to Automated Theorem Proving. • Proving by instances. • Given a triangle abc and a point d on its circumcircle, the feet e, f, g of the perpendi- culars from d to the lines bc , ab , and ac , respectively, are collinear. f=(t,u) d=(m,n) b=(r,s) e=(v,w) c=(0,0) a=(1,0) g=(m,0) Figure 1: Illustration of Simson’s Theorem

  8. S8 Goal I: Test Set. Definition 1 We recall that the geometric degree 2 of an irreducible affine variety U ⊂ K k is the number of intersections of U with a generic affine linear variety of codimension dim( U ) . When the variety is reducible, the degree is defined as the sum of the degrees of the irreducible components. Figure 2: Illustration of geometric degree 2 Heintz J. (1983). Definability and fast quantifier elimination in algebraically closed fields. Theoretical Computer Science , 24, pp. 239-277.

  9. S9 Goal I: Test Set. Definition 2 (TEST SET) A finite subset A ⊆ K r is a TEST SET for the varieties of geometric degree less or equal than d with d > 0 , (shortly a ( d, r ) -test set ), if no proper variety W ⊂ K r of gdeg( W ) ≤ d contains A . Theorem 1 Let A ⊆ K r and d ∈ Z > 0 . Then A is a ( d, r ) -test set if and only if no hypersurface of K r , of geometric degree less or equal than d , contains A . Figure 3: Illustration of test sets

  10. S10 Goal I: Supp(d,r). For d, r ∈ Z > 0 , we denote by r � Supp( d, r ) = { ( x 1 , . . . , x r ) ⊆ Z r ≥ 0 | x i ≤ d } i =1 the set of exponents on the support of a generic polynomial of degree d in r variables. • Supp(2 , 2) = { (0 , 0) , (0 , 1) , (1 , 0) , (1 , 1) , (0 , 2) , (2 , 0) } . • Supp(2 , 3) = { (0 , 0 , 0) , (0 , 0 , 1) , (0 , 1 , 0) , (0 , 1 , 1) , (1 , 0 , 0) , (1 , 0 , 1) , (1 , 1 , 0) , (1 , 1 , 1) , (2 , 0 , 0) , (0 , 2 , 0) , (0 , 0 , 2) } � d + r � • #(Supp( d, r )) = . r

  11. S11 Goal I: Supp(d,r). Theorem 2 Supp( d, r ) is a ( d, r ) -test set of minimum cardinality. Figure 4: Illustration of different supports

  12. S12 Goal I: Test Set. Theorem 3 Let A be a ( d, r ) -test set, and ϕ a bijective affine transformation of K r . Then ϕ ( A ) is a ( d, r ) -test set. � x � 1 − 1 � � x � 1 � � � ϕ = + 1 y 1 1 y 2

  13. S13 Goal I: Disjuntive Test Set. In our context (Goals II and III) it is required to construct sets having stronger properties than that of being a test set, namely, such that any subset of cardinal greater than a fixed size is also a test set. Definition 3 Let d, r ∈ Z > 0 , and N = #(Supp( d, r )) . We say that a finite set A , with #( A ) ≥ N , is a ( d, r ) -disjunctive test set if any subset of A of cardinal N is a ( d, r ) -test set. The motivation of this notion is the following. Assume that A is disjunctive and #( A ) ≥ 2 N − 1 and B ⊆ A , then either B or A \ B is a ( d, r ) -test set. Indeed, if #( B ) ≥ N , the statement holds by the definition of disjunctive test set. Else, #( A \ B ) ≥ N , and thus A \ B is a ( d, r ) -test set.

  14. S14 Goal I: Disjuntive Test Set. Algorithm 1. Given d, r ∈ Z ≥ 0 , the algorithm derives a ( d, r ) -disjunctive test set of any given cardinal M ≥ N = #(Supp( d, r )) . 1. If M = N Return Supp( d, r ) . 2. Set B = Supp( d, r ) . 3. For i from 1 to M − N do (a) For any subset C of B with #( C ) = N − 1 determine the unique hypersurface H C of K r of degree d . passing through C. (b) Compute a point P ∈ K r not in any of the hypersurfaces obtained in the previous step. (c) Set B = B ∪ { P } . 4. Return B .

  15. S15 Goal I: Disjuntive Test Set. Example 1 (2 , 2) -disjunctive test set of cardinal 7. 1. Let us consider N = #(Supp(2 , 2)) = 6 and let M = 7 . 2. Supp(2 , 2) = { P 1 , P 2 , P 3 , P 4 , P 5 , P 6 } = { (0 , 0) , (1 , 0) , (2 , 0) , (0 , 1) , (1 , 1) , (0 , 2) } . 3. Let H i be the unique conic passing through Supp(2 , 2) \ { P i } . H 1 = ( x + y − 2)( x + y − 1) H 2 = x ( x + y − 2) H 3 = x ( x − 1) H 4 = y ( x + y − 2) H 5 = xy H 6 = y ( y − 1) 4. Then taking P / ∈ ∪ V ( H i ) , for instance, P = (2 / 3 , 2 / 3) . 5. Supp(2 , 2) ∪ { P } is a (2 , 2) -disjunctive test set of cardinal 7.

  16. S16 Goal II: Testing the Nullity of Elimination Ideals. Figure 5: Illustration of the proyection. If gdeg( V ) = d then gdeg( V r ) ≤ d 3 and gdeg( V r \ π r ( V )) ≤ d 4 . 3 Heintz J. (1983). Definability and fast quantifier elimination in algebraically closed fields. Theoretical Computer Science , 24, pp. 239-277. 4 Personal communication by prof. Mart´ ın Sombra, ICREA Research Professor at Universitat de Barcelona, Spain.

  17. S17 Goal II: Testing the Nullity of Elimination Ideals. Figure 6: Illustration of the proyection.

  18. S18 Goal II: Testing the Nullity of Elimination Ideals. Figure 7: Illustration of the proyection.

  19. S19 Goal II: Testing the Nullity of Elimination Ideals. Algorithm 2. Given a bound d for the geometric degree of V , the algorithm decides whether the ideal I r is zero or not. 1. Set N = ( d + r r ) . 2. Apply Algorithm 1 to N and ( d, r ) to get a ( d, r ) –disjunctive test set of cardinality 2 N − 1 , say C . 3. Using an oracle , decompose C as C = A ∪ B , where for every P ∈ A it holds that P ∈ π r ( V ) and for every P ∈ B it holds that P �∈ π r ( V ) 4. If #( A ) ≥ N then Return I r = < 0 > else I r � = < 0 > .

  20. S20 Goal II: Testing the Nullity of Elimination Ideals. Algorithm 2. Given a bound d for the geometric degree of V , the algorithm decides whether the ideal I r is zero or not. 1. Set N = ( d + r r ) . 2. Apply Alg. 1 to ( d, r ) to get a ( d, r ) –disjunctive test set C of cardinality 2 N − 1 . • One may have a pre-computed data basis C , for different values of d and r . • If not one may combine Alg. 1 and 2: whenever a point P ∈ C is computed, one decides whether P belongs or not to π r ( V ) . As soon as the cardinality of either A or B is greater or equal N , the process can be stopped.... • A third option, probably the most efficient, is to compute T = Supp( d, r ) and to apply a random linear transformation to T (see Theorem 3) to get T ∗ . In this situation, we check how many points in T ∗ can be lifted to V .... 3. Using an oracle .... 4. If ...

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