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Polynomial Interpolation: an Introduction to Algebraic Geometry

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Polynomial Interpolation: an Introduction to Algebraic Geometry POLYNOMIAL INTERPOLATION: AN INTRODUCTION TO ALGEBRAIC GEOMETRY JACK HUIZENGA Abstract. These are notes for the Fall 2018 Honors MASS Al- gebra course at Penn State. They cover a one-semester advanced undergraduate course in linear algebra and algebraic geometry. Contents 1. Lagrangian Interpolation 1 2. Linear Algebra: Vector Spaces 11 3. Linear Algebra: Linear Transformations 33 4. Multivariate interpolation: the Hilbert function 43 5. Algebraic geometry: varieties in affine space 50 6. Parameter spaces 69 7. The Alexander-Hirschowitz theorem 85 8. Exercises 97 9. Topics for further study 109 1. Lagrangian Interpolation First let’s fix some notation. Let x1, . , xn R be distinct points, ∈ and let y1, . , yn R be arbitrary values. The interpolation problem ∈ for this data is to find a function f : R R which takes the prescribed values at the points: → f(xi) = yi (i = 1, . , n). Already in high school algebra, students want to solve the interpola- tion problem. How can we find a function which takes the prescribed values at the prescribed points? On the face of it this question is not really very interesting: a function f : R R is an arbitrary assignment → of a value f(x) R for each x R, so we can for instance construct a ∈ ∈ Date: January 24, 2019. 1 2 J. HUIZENGA function which has f(xi) = yi for all i by defining y if x = x f(x) = i i (0 otherwise. There is nothing particular about the value 0 here; we can instead (independently) assign whatever value we like at any points x different from x1, . , xn. The set of all functions f : R R is enormous, and contains all kinds of nasty functions that no high→ schooler would be happy with. Their graphs look like the static fuzz of an old TV screen. Of course what the high schooler really wants to do is come up with a “nice,” “named” function f : R R that satisfies f(xi) = yi. They don’t want an f defined piecewise,→ and hopefully f is differentiable. For the interpolation problem to be interesting, we need to restrict the class of functions that we are considering. The simplest, most natural family of functions to look at are the polynomial functions. 1.1. Basics on polynomials, functions, and polynomial func- tions. Definition 1.1 (Polynomials). A polynomial p (with real coefficients) is an expression of the form p = a xn + + a x + a n ∙ ∙ ∙ 1 0 where the ai R. ∈ The set of all such polynomials is denoted by R[x]. We can add poly- n i m i nomials coefficient-by-coefficient: if p = i=0 aix and q = i=0 bix , then max m,n P P { } i p + q = (ai + bi)x . i=0 X The product of p and q is the polynomial m+n k pq = ckx with ck = aibj. Xk=0 i+Xj=k With these operations, the set R[x] is what is called a ring in algebra. (Basically, this just means that the usual rules you are accustomed to for addition and multiplication hold. For example, p(q + r) = pq + pr.) Two polynomials p and q are equal iff they are given by the same expression. In other words, their coefficients are equal. Definition 1.2 (Functions). Let X and Y be sets. A function f : X Y assigns to each x X an element f(x) Y . Two functions → ∈ ∈ POLYNOMIAL INTERPOLATION 3 f, g : X Y are equal iff f(x) = g(x) for all x X. (This is just set theory.) → ∈ Definition 1.3 (Real-valued functions). Let X be a set and consider functions f : X R. (A particularly interesting case is where also → X = R.) The set of all such functions is denoted by RX . We can use the addition and multiplication operations on R to define addition and multiplication operations on RX . Specifically, if f, g : X R then we define functions f + g and fg pointwise by the rules → (f + g)(x) := f(x) + g(x) (fg)(x) := f(x)g(x). With these operations, the set RX again becomes an example of a ring. Definition 1.4 (Polynomial functions). Given a polynomial p R[x], ∈ we can substitute a real value t R for the variable x. Thus if p = i ∈ aix , we can define a function f : R R by the rule → i P f(t) = ait . The function f is called a polynomialX function. It can be represented by the polynomial p. The set of all polynomial functions is a subset of RR, the set of all functions from R to R. Warning 1.5. The polynomial p representing a polynomial function f : R R is unique, but this requires proof. (See below.) In particular, → two polynomial functions f, g : R R are equal if and only if the polynomials p, q representing them→ are equal. Polynomial functions f : R R and polynomials p R[x] are “basically” the same thing, but in→ more general contexts this∈ requires some care. n Definition 1.6 (Degree). The degree of the polynomial p = anx + + a0 is deg p = n if an = 0. We define deg 0= by convention. The∙ ∙ ∙ degree of a polynomial6 function is the degree of−∞ a polynomial which represents it. Lemma 1.7 (Basic properties of degree). Let p, q R[x]. ∈ (1) We have deg(pq) = deg p + deg q. (2) We have deg(p + q) max deg p, deg q . If deg p = deg q, then equality holds. ≤ { } 6 1.2. Lagrangian interpolation: existence. Back to our motivating question. Let x1, . , xn R be distinct points, and let y1, . , yn R be arbitrary values. How∈ can we construct a polynomial function∈f : R R such that f(xi) = yi for all i? → 4 J. HUIZENGA First, let’s solve a special case of this problem. What if y1 = 1 and all other yi = 0? We can easily arrange that a polynomial function f(x) has f(xi) = 0 by giving it a factor of (x xi). Thus, the polynomial function − f(x) = (x x ) (x x ) − 2 ∙ ∙ ∙ − n has f(x ) = 0 for i = 2, . , n, and f(x ) = 0. To get a value of 1 at i 1 6 x1, we can just divide by the constant f(x1). Therefore (x x ) (x x ) F (x) = − 2 ∙ ∙ ∙ − n 1 (x x ) (x x ) 1 − 2 ∙ ∙ ∙ 1 − n is a polynomial function such that F (x ) = 1 and F (x ) = 0 for i = 1. 1 1 1 i 6 Similarly, we can define for each i a polynomial function Fi(x) by the rule n (x xj) Fi(x) = − . j=1 (xi xj) j=i − Y6 Then 1 i = j F (x ) = δ := i j ij 0 i = j. ( 6 We can piece these special solutions together to prove the following theorem. Theorem 1.8 (Lagrangian interpolation: existence). Let x , . , x 1 n ∈ R be distinct points, and let y1, . , yn R be arbitrary values. Then ∈ there exists a polynomial function f : R R of degree at most n 1 → − such that f(xi) = yi. Proof. In the notation preceding the statement, the function n yiFi i=1 X has the required properties. The basic method of proof used here was to first find a polynomial that vanishes at all but one of the points, to construct a “characteristic function” supported at just one of the points. Then it is a simple matter to combine these various functions to get whatever value we want at each of the points. This technique continues to be useful in more general problems, as we will see. Could there have been other solutions to the interpolation problem? If we don’t bound the degree of the function, then yes. Notice that the POLYNOMIAL INTERPOLATION 5 polynomial (x x1) (x xn) vanishes at all the points x1, . , xn. Therefore, if f(−x) is any∙ ∙ ∙ solution− to the interpolation problem, then f(x) + (x x ) (x x ) − 1 ∙ ∙ ∙ − n is also a solution. More generally, if g is any polynomial, then f(x) + (x x ) (x x )g(x) − 1 ∙ ∙ ∙ − n is also a solution. However, if deg f n 1, then all these other solutions have degree n or greater, so≤ we have− only constructed one solution of degree at most n 1. − 1.3. The division algorithm. It can’t be overstated how important the division algorithm is in the study of 1-variable polynomials. In algebra terminology, R[x] is what is called a Euclidean domain, and it is one of the simplest possible kinds of rings out there. This fact is directly responsible for the simple story of Lagrangian interpolation. Theorem 1.9 (The division algorithm). Let p, m R[x] with m = ∈ 6 0. Then there are unique polynomials q, r R[x] (the quotient and remainder) with deg r < deg m such that ∈ p = qm + r. The proof of the theorem is essentially an analysis of the high school algorithm for polynomial long division. Proof. (Existence) First we show that we can find such polynomials q and r. We go by (strong) induction on the degree of p. If deg p < deg m we take q = 0 and r = p. Suppose deg p deg m and that the result ≥ is known for polynomials p0 of degree deg p0 < deg p. Since deg p deg m and m = 0, there is a monomial cxa such that ≥ 6 a p0 := p cx m − has degree < deg p. (Choose c and a to cancel the leading term of p.) Then by induction we can write p0 = q0m + r0 (deg r0 < deg m).
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