{"id":356468,"date":"2017-01-21T05:55:07","date_gmt":"2017-01-21T13:55:07","guid":{"rendered":"https:\/\/cm-edgetun.pages.dev\/en-us\/research\/?post_type=msr-research-item&#038;p=356468"},"modified":"2018-10-16T20:55:11","modified_gmt":"2018-10-17T03:55:11","slug":"restrictions-nondegenerate-boolean-functions-degree-lower-bounds-different-rings","status":"publish","type":"msr-research-item","link":"https:\/\/cm-edgetun.pages.dev\/en-us\/research\/publication\/restrictions-nondegenerate-boolean-functions-degree-lower-bounds-different-rings\/","title":{"rendered":"Restrictions of nondegenerate Boolean functions and degree lower bounds over different rings"},"content":{"rendered":"<p>A Boolean function f : {0, 1}<sup>n<\/sup> \u2192 {0, 1} is called nondegenerate if f depends on all its n variables. We show that, for any nondegenerate function f, there exists a variable x<sub>i<\/sub> such that at least one of the restrictions f<sub>Ix<\/sub><sub>i<\/sub><sub>=0<\/sub> or f<sub>Ix<\/sub><sub>i<\/sub><sub>=1<\/sub> must depend on all the remaining n &#8211; 1 variables. We also consider lower bounds on the degrees of polynomials representing a Boolean function over different rings. Let d<sub>q<\/sub>(f) be the degree of the (unique) polynomial over the ring \u2124<sub>q<\/sub> exactly representing f. For distinct primes p<sub>i<\/sub> let m = \u03a0<sup>r<\/sup><sub>i=1<\/sub> p<sup>ei<\/sup><sub>i<\/sub>. Then, we show that any nondegenerate symmetric Boolean function f must have m \u00b7 d<sub>p<\/sub><sub>1<\/sub>e<sub>1<\/sub>(f)&#8230;d<sub>pr<\/sub>e<sub>r<\/sub>(f) > n. We use the existence of nondegenerate subfunctions to prove degree lower bounds on random functions. Specifically, we show that m \u00b7 d<sub>p<\/sub><sub>1<\/sub>e<sub>1<\/sub>(f)&#8230;d<sub>p<\/sub><sub>r<\/sub>e<sub>r<\/sub>(f) > lg n &#8211; 1 holds for almost all f when f is chosen uniformly at random from all n-variate Boolean functions. Our proof uses the second moment method to show that a random f must almost always contain a nondegenerate symmetric subfunction on at least lg n &#8211; 1 variables. It follows that an n-variate nondegenerate symmetric Boolean function can have degree o(\u221a(n)) over at most one finite field and that almost all f can have degree o(\u221a(lg n)) over at most one finite field.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>A Boolean function f : {0, 1}n \u2192 {0, 1} is called nondegenerate if f depends on all its n variables. We show that, for any nondegenerate function f, there exists a variable xi such that at least one of the restrictions fIxi=0 or fIxi=1 must depend on all the remaining n &#8211; 1 variables. 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