PUMaC 2014 · 加试 · 第 4 题
PUMaC 2014 — Power Round — Problem 4
题目详情
- (multiplicity) = p p p Theorem 5 (Quadratic Reciprocity) . Given two odd primes p 6 = q , we have ( ) ( ) ( p − 1)( q − 1) p q 4 = ( − 1) . q p ( ) ( ) p q Stated differently, = unless p ≡ q ≡ 3 (mod 4) . q p 2.2 Background for This Year’s Power Round This year’s Power Round concerns the following interesting mathematical ob- ject. 3 Definition 6 (Conic Polynomial) . A Conic Polynomial involving n variables X , X , · · · , X is the homogeneous polynomial 1 2 n ∑ f = f ( X , X , · · · , X ) = a X X 1 2 n i,j i j 1 ≤ i,j ≤ n where a are real numbers and the sum ranges over all pairs ( i, j ) with i,j 1 ≤ i ≤ n and 1 ≤ j ≤ n . We may write this conic polynomial in the following matrix notation : a a · · · a X 1 , 1 1 , 2 1 ,n 1 a a · · · a X ( ) 2 , 1 2 , 2 2 ,n 2 T f = X AX = X X · · · X . . . . 1 2 n . . . . . . . . . . . a a · · · a X n, 1 n, 2 n,n n We say that a conic polynomial f is integral if for all integers X , X , · · · , X , 1 2 n f ( X , X , · · · , X ) is also an integer. 1 2 n 2 2 2 2 We can easily see that the forms x + 3 xy + 5 y , x − y are two-variable 2 2 2 integral conic polynomials, and x + y + z , xy + xz are three-variable integral conic polynomials. Definition 7. Multiple matrices may be associated with the same conic poly- 2 2 nomial. For example, the conic polynomial x + 4 xy + y can be associated with the two matrices ( ) 1 3 A = 1 1 1 and ( ) 1 4 A = . 2 0 1 However, there is a unique way to associate a conic polynomial with a symmetric 2 2 matrix A = ( a ) with a = a . For example, x + 4 xy + y is associated to i,j i,j j,i the following matrix ( ) 1 2 A = . 2 1 We call this matrix the symmetric matrix associated to a conic polynomial f . If the symmetric matrix associated to a conic polynomial f has integer entries, we say that f has integer matrix . Definition 8. We say that a conic polynomial f ( X , X , · · · , X ) is integral if 1 2 n f ( X , X , · · · , X ) is an integer for all integers X , X , · · · , X . Note that this 1 2 n 1 2 n is not equivalent to the fact that f has integer matrix. Definition 9. We say that a conic polynomial f ( X , X , · · · , X ) represents 1 2 n an integer d ∈ Z if f ( X , X , · · · , X ) = d has a solution with X ∈ Z for all i . 1 2 n i 4 Definition 10. We say that a conic polynomial f is positive-definite if f ≥ 0 for all integer inputs and f = 0 iff all arguments are zero. Definition 11. We say that a positive-definite conic polynomial f is universal if f represents all nonnegative integers. Let’s begin by gaining some intuition on these conic polynomial, and get used to the definitions.
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