Given a binary quadratic polynomial f(x_(1),x_(2))=αx_(1)^(2)+βx_(1)x_(2)+γx_(2)^(2)∈Z[x_(1),x_(2)],for every c∈Z and n≥2,we study the number of solutions NJ(f;c,n)of the congruence equation f(x_(1),x_(2))≡c mo...Given a binary quadratic polynomial f(x_(1),x_(2))=αx_(1)^(2)+βx_(1)x_(2)+γx_(2)^(2)∈Z[x_(1),x_(2)],for every c∈Z and n≥2,we study the number of solutions NJ(f;c,n)of the congruence equation f(x_(1),x_(2))≡c mod n in(Z/nZ)^(2) such that xi∈(Z/nZ)^(×)for i∈J⊆{1,2}.展开更多
Kapur and Musser studied the theoretical basis for proof by consistency and ob-tained an inductive completeness result: p q if and only if p = q is true in everyinductive model. However, there is a loophole in their p...Kapur and Musser studied the theoretical basis for proof by consistency and ob-tained an inductive completeness result: p q if and only if p = q is true in everyinductive model. However, there is a loophole in their proof for the soundness part:p = q implies p = q is true in every inductive model. The aim of this paper is to give acorrect characterization of inductive soundness from an algebraic view by introducingstrong inductive models.展开更多
基金supported by National Natural Science Foundation of China(Grant No.11571328).
文摘Given a binary quadratic polynomial f(x_(1),x_(2))=αx_(1)^(2)+βx_(1)x_(2)+γx_(2)^(2)∈Z[x_(1),x_(2)],for every c∈Z and n≥2,we study the number of solutions NJ(f;c,n)of the congruence equation f(x_(1),x_(2))≡c mod n in(Z/nZ)^(2) such that xi∈(Z/nZ)^(×)for i∈J⊆{1,2}.
文摘Kapur and Musser studied the theoretical basis for proof by consistency and ob-tained an inductive completeness result: p q if and only if p = q is true in everyinductive model. However, there is a loophole in their proof for the soundness part:p = q implies p = q is true in every inductive model. The aim of this paper is to give acorrect characterization of inductive soundness from an algebraic view by introducingstrong inductive models.
