Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

• Username
• Password
• Remember me

• Sign in using OpenID
• Remember me

• Forgot your password?
• Apply for membership

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory internal-categories k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

1. • CommentRowNumber1.
   • CommentAuthorTodd_Trimble
   • CommentTimeSep 6th 2015
   • (edited Sep 6th 2015)
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexI started an article [[bijective proof]]. In a section on polynomial identities, I give a proof that polynomials in several variables are uniquely determined by their values at natural number arguments, an intuitively obvious statement if there ever was one. Without bothering to look up whether there are standard nice proofs, I cooked up a proof myself. Please let me know if you know of nicer proofs. Edit: Having written this, it's painfully obvious how to prove more general statements even more simply. Ah well. I still invite comments.

   I started an article bijective proof.

   In a section on polynomial identities, I give a proof that polynomials in several variables are uniquely determined by their values at natural number arguments, an intuitively obvious statement if there ever was one. Without bothering to look up whether there are standard nice proofs, I cooked up a proof myself. Please let me know if you know of nicer proofs.

   Edit: Having written this, it’s painfully obvious how to prove more general statements even more simply. Ah well. I still invite comments.

2. • CommentRowNumber2.
   • CommentAuthorZhen Lin
   • CommentTimeSep 6th 2015
   • PermaLink
   Author: Zhen Lin
   Format: MarkdownItexIt suffices to prove the following: > Let $A$ be an infinite integral domain and let $B$ be an infinite subset of $A$. Given $p (x_1, \ldots, x_n) \in A [x_1, \ldots, x_n]$, if $p (b_1, \ldots, b_n) = 0$ for all $(b_1, \ldots, b_n) \in B^n$, then $p = 0$. We proceed by induction on $n$. The case $n = 1$ is well known (at least if $A$ is an algebraically closed field – but we can embed an integral domain in the algebraic closure of its fraction field). If the claim holds for $n = k$, then given $p (x_1, \ldots, x_{k+1}) \in A [x_1, \ldots, x_{k+1}]$ such that $p (b_1, \ldots, b_{k+1}) = 0$ for all $(b_1, \ldots, b_{k+1}) \in B^{k+1}$, we know that $p (x_1, \ldots, x_k, b)$ is the zero polynomial for all $b \in B$; but then $p (x_1, \ldots, x_k, x_{k+1})$ is a one-variable polynomial over $A [x_1, \ldots, x_k]$ that vanishes at infinitely many points, so $p = 0$.

   It suffices to prove the following:

   Let $A$ be an infinite integral domain and let $B$ be an infinite subset of $A$. Given $p (x_1, \ldots, x_n) \in A [x_1, \ldots, x_n]$, if $p (b_1, \ldots, b_n) = 0$ for all $(b_1, \ldots, b_n) \in B^n$, then $p = 0$.

   We proceed by induction on $n$. The case $n = 1$ is well known (at least if $A$ is an algebraically closed field – but we can embed an integral domain in the algebraic closure of its fraction field). If the claim holds for $n = k$, then given $p (x_1, \ldots, x_{k+1}) \in A [x_1, \ldots, x_{k+1}]$ such that $p (b_1, \ldots, b_{k+1}) = 0$ for all $(b_1, \ldots, b_{k+1}) \in B^{k+1}$, we know that $p (x_1, \ldots, x_k, b)$ is the zero polynomial for all $b \in B$; but then $p (x_1, \ldots, x_k, x_{k+1})$ is a one-variable polynomial over $A [x_1, \ldots, x_k]$ that vanishes at infinitely many points, so $p = 0$.

3. • CommentRowNumber3.
   • CommentAuthorTodd_Trimble
   • CommentTimeSep 6th 2015
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexYep, thanks. I would have added the (obvious) remark that we first embed $A[x_1, \ldots, x_k]$ into its field of fractions $K$, and then use well-known facts about $K[x]$.

   Yep, thanks. I would have added the (obvious) remark that we first embed $A[x_1, \ldots, x_k]$ into its field of fractions $K$, and then use well-known facts about $K[x]$.

4. • CommentRowNumber4.
   • CommentAuthorTodd_Trimble
   • CommentTimeSep 6th 2015
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexSo I went ahead and supplanted the earlier clunky proof with a version of Zhen Lin's (thanks again).

   So I went ahead and supplanted the earlier clunky proof with a version of Zhen Lin’s (thanks again).

5. • CommentRowNumber5.
   • CommentAuthorDavidRoberts
   • CommentTimeSep 6th 2015
   • PermaLink
   Author: DavidRoberts
   Format: MarkdownItexDoron Zeilberger is fond of proving identities by checking them at the smallest few nontrivial inputs, eg 1,2 and 3, where these are trivial to calculate.

   Doron Zeilberger is fond of proving identities by checking them at the smallest few nontrivial inputs, eg 1,2 and 3, where these are trivial to calculate.

6. • CommentRowNumber6.
   • CommentAuthorTodd_Trimble
   • CommentTimeApr 21st 2016
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexI added some details to some examples that had been there. Will probably write (and rewrite) more later.

   I added some details to some examples that had been there. Will probably write (and rewrite) more later.